Adding upstream version 16.2.11+ds.upstream/16.2.11+dsupstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 2565 insertions, 0 deletions

diff --git a/src/include/cpp-btree/btree.h b/src/include/cpp-btree/btree.h
new file mode 100644
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+++ b/src/include/cpp-btree/btree.h
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+// Copyright 2018 The Abseil Authors.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//      https://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+// A btree implementation of the STL set and map interfaces. A btree is smaller
+// and generally also faster than STL set/map (refer to the benchmarks below).
+// The red-black tree implementation of STL set/map has an overhead of 3
+// pointers (left, right and parent) plus the node color information for each
+// stored value. So a set<int32_t> consumes 40 bytes for each value stored in
+// 64-bit mode. This btree implementation stores multiple values on fixed
+// size nodes (usually 256 bytes) and doesn't store child pointers for leaf
+// nodes. The result is that a btree_set<int32_t> may use much less memory per
+// stored value. For the random insertion benchmark in btree_bench.cc, a
+// btree_set<int32_t> with node-size of 256 uses 5.1 bytes per stored value.
+//
+// The packing of multiple values on to each node of a btree has another effect
+// besides better space utilization: better cache locality due to fewer cache
+// lines being accessed. Better cache locality translates into faster
+// operations.
+//
+// CAVEATS
+//
+// Insertions and deletions on a btree can cause splitting, merging or
+// rebalancing of btree nodes. And even without these operations, insertions
+// and deletions on a btree will move values around within a node. In both
+// cases, the result is that insertions and deletions can invalidate iterators
+// pointing to values other than the one being inserted/deleted. Therefore, this
+// container does not provide pointer stability. This is notably different from
+// STL set/map which takes care to not invalidate iterators on insert/erase
+// except, of course, for iterators pointing to the value being erased.  A
+// partial workaround when erasing is available: erase() returns an iterator
+// pointing to the item just after the one that was erased (or end() if none
+// exists).
+
+#pragma once
+
+#include <algorithm>
+#include <cassert>
+#include <cstddef>
+#include <cstdint>
+#include <cstring>
+#include <experimental/type_traits>
+#include <functional>
+#include <iterator>
+#include <limits>
+#include <new>
+#include <type_traits>
+#include <utility>
+
+namespace btree::internal {
+
+template <typename Compare, typename T>
+using btree_is_key_compare_to =
+  std::is_signed<std::invoke_result_t<Compare, T, T>>;
+
+template<typename T>
+using compare_to_t = decltype(std::declval<T&>().compare(std::declval<const T&>()));
+template<typename T>
+inline constexpr bool has_compare_to = std::experimental::is_detected_v<compare_to_t, T>;
+// A helper class to convert a boolean comparison into a three-way "compare-to"
+// comparison that returns a negative value to indicate less-than, zero to
+// indicate equality and a positive value to indicate greater-than. This helper
+// class is specialized for less<std::string>, greater<std::string>,
+// less<string_view>, and greater<string_view>.
+//
+// key_compare_to_adapter is provided so that btree users
+// automatically get the more efficient compare-to code when using common
+// google string types with common comparison functors.
+// These string-like specializations also turn on heterogeneous lookup by
+// default.
+template <typename Compare, typename=void>
+struct key_compare_to_adapter {
+  using type = Compare;
+};
+
+template <typename K>
+struct key_compare_to_adapter<std::less<K>, std::enable_if_t<has_compare_to<K>>>
+{
+  struct type {
+    inline int operator()(const K& lhs, const K& rhs) const noexcept {
+      return lhs.compare(rhs);
+    }
+  };
+};
+
+template <typename K>
+struct key_compare_to_adapter<std::less<K>, std::enable_if_t<std::is_signed_v<K>>>
+{
+  struct type {
+    inline K operator()(const K& lhs, const K& rhs) const noexcept {
+      return lhs - rhs;
+    }
+  };
+};
+
+template <typename K>
+struct key_compare_to_adapter<std::less<K>, std::enable_if_t<std::is_unsigned_v<K>>>
+{
+  struct type {
+    inline int operator()(const K& lhs, const K& rhs) const noexcept  {
+      if (lhs < rhs) {
+        return -1;
+      } else if (lhs > rhs) {
+        return 1;
+      } else {
+        return 0;
+      }
+    }
+  };
+};
+
+template <typename Key, typename Compare, typename Alloc,
+          int TargetNodeSize, int ValueSize,
+          bool Multi>
+struct common_params {
+  // If Compare is a common comparator for a std::string-like type, then we adapt it
+  // to use heterogeneous lookup and to be a key-compare-to comparator.
+  using key_compare = typename key_compare_to_adapter<Compare>::type;
+  // A type which indicates if we have a key-compare-to functor or a plain old
+  // key-compare functor.
+  using is_key_compare_to = btree_is_key_compare_to<key_compare, Key>;
+
+  using allocator_type = Alloc;
+  using key_type = Key;
+  using size_type = std::make_signed<size_t>::type;
+  using difference_type = ptrdiff_t;
+
+  // True if this is a multiset or multimap.
+  using is_multi_container = std::integral_constant<bool, Multi>;
+
+  constexpr static int kTargetNodeSize = TargetNodeSize;
+  constexpr static int kValueSize = ValueSize;
+  // Upper bound for the available space for values. This is largest for leaf
+  // nodes, which have overhead of at least a pointer + 3 bytes (for storing
+  // 3 field_types) + paddings. if alignof(key_type) is 1, the size of padding
+  // would be 0.
+  constexpr static int kNodeValueSpace =
+        TargetNodeSize - /*minimum overhead=*/(sizeof(void *) + 4);
+
+  // This is an integral type large enough to hold as many
+  // ValueSize-values as will fit a node of TargetNodeSize bytes.
+  using node_count_type =
+      std::conditional_t<(kNodeValueSpace / ValueSize >
+                          (std::numeric_limits<uint8_t>::max)()),
+                         uint16_t,
+                         uint8_t>;
+};
+
+// The internal storage type
+//
+// It is convenient for the value_type of a btree_map<K, V> to be
+// pair<const K, V>; the "const K" prevents accidental modification of the key
+// when dealing with the reference returned from find() and similar methods.
+// However, this creates other problems; we want to be able to emplace(K, V)
+// efficiently with move operations, and similarly be able to move a
+// pair<K, V> in insert().
+//
+// The solution is this union, which aliases the const and non-const versions
+// of the pair. This also allows flat_hash_map<const K, V> to work, even though
+// that has the same efficiency issues with move in emplace() and insert() -
+// but people do it anyway.
+template <class K, class V>
+union map_slot_type {
+  map_slot_type() {}
+  ~map_slot_type() = delete;
+  map_slot_type& operator=(const map_slot_type& slot) {
+    mutable_value = slot.mutable_value;
+    return *this;
+  }
+  map_slot_type& operator=(map_slot_type&& slot) {
+    mutable_value = std::move(slot.mutable_value);
+    return *this;
+  }
+  using value_type = std::pair<const K, V>;
+  using mutable_value_type = std::pair<K, V>;
+
+  value_type value;
+  mutable_value_type mutable_value;
+  K key;
+};
+
+template <class K, class V>
+void swap(map_slot_type<K, V>& lhs, map_slot_type<K, V>& rhs) {
+  std::swap(lhs.mutable_value, rhs.mutable_value);
+}
+
+// A parameters structure for holding the type parameters for a btree_map.
+// Compare and Alloc should be nothrow copy-constructible.
+template <typename Key, typename Data, typename Compare, typename Alloc,
+          int TargetNodeSize, bool Multi>
+struct map_params : common_params<Key, Compare, Alloc, TargetNodeSize,
+                                  sizeof(Key) + sizeof(Data), Multi> {
+  using super_type = typename map_params::common_params;
+  using mapped_type = Data;
+  using value_type = std::pair<const Key, mapped_type>;
+  using mutable_value_type = std::pair<Key, mapped_type>;
+  using slot_type = map_slot_type<Key, mapped_type>;
+  using pointer = value_type*;
+  using const_pointer = const value_type *;
+  using reference = value_type &;
+  using const_reference = const value_type &;
+  using key_compare = typename super_type::key_compare;
+  using init_type = mutable_value_type;
+
+  static constexpr size_t kValueSize = sizeof(Key) + sizeof(mapped_type);
+
+  // Inherit from key_compare for empty base class optimization.
+  struct value_compare : private key_compare {
+    value_compare() = default;
+    explicit value_compare(const key_compare &cmp) : key_compare(cmp) {}
+
+    template <typename T, typename U>
+    auto operator()(const T &left, const U &right) const
+        -> decltype(std::declval<key_compare>()(left.first, right.first)) {
+      return key_compare::operator()(left.first, right.first);
+    }
+  };
+  using is_map_container = std::true_type;
+
+  static const Key &key(const value_type &value) { return value.first; }
+  static mapped_type &value(value_type *value) { return value->second; }
+  static const Key &key(const slot_type *slot) { return slot->key; }
+  static value_type& element(slot_type* slot) { return slot->value; }
+  static const value_type& element(const slot_type* slot) { return slot->value; }
+  template <class... Args>
+  static void construct(Alloc *alloc, slot_type *slot, Args &&... args) {
+    std::allocator_traits<Alloc>::construct(*alloc,
+                                            &slot->mutable_value,
+                                            std::forward<Args>(args)...);
+  }
+  // Construct this slot by moving from another slot.
+  static void construct(Alloc* alloc, slot_type* slot, slot_type* other) {
+    emplace(slot);
+    std::allocator_traits<Alloc>::construct(*alloc, &slot->value,
+                                            std::move(other->value));
+  }
+  static void move(Alloc *alloc, slot_type *src, slot_type *dest) {
+    dest->mutable_value = std::move(src->mutable_value);
+  }
+  static void destroy(Alloc *alloc, slot_type *slot) {
+    std::allocator_traits<Alloc>::destroy(*alloc, &slot->mutable_value);
+  }
+
+private:
+  static void emplace(slot_type* slot) {
+    // The construction of union doesn't do anything at runtime but it allows us
+    // to access its members without violating aliasing rules.
+    new (slot) slot_type;
+  }
+};
+
+// A parameters structure for holding the type parameters for a btree_set.
+template <typename Key, typename Compare, typename Alloc, int TargetNodeSize, bool Multi>
+struct set_params
+    : public common_params<Key, Compare, Alloc, TargetNodeSize,
+                           sizeof(Key), Multi> {
+  using value_type = Key;
+  using mutable_value_type = value_type;
+  using slot_type = Key;
+  using pointer = value_type *;
+  using const_pointer = const value_type *;
+  using value_compare = typename set_params::common_params::key_compare;
+  using reference = value_type &;
+  using const_reference = const value_type &;
+  using is_map_container = std::false_type;
+  using init_type = mutable_value_type;
+
+  template <class... Args>
+  static void construct(Alloc *alloc, slot_type *slot, Args &&... args) {
+    std::allocator_traits<Alloc>::construct(*alloc,
+                                            slot,
+                                            std::forward<Args>(args)...);
+  }
+  static void construct(Alloc *alloc, slot_type *slot, slot_type *other) {
+    std::allocator_traits<Alloc>::construct(*alloc, slot, std::move(*other));
+  }
+  static void move(Alloc *alloc, slot_type *src, slot_type *dest) {
+    *dest = std::move(*src);
+  }
+  static void destroy(Alloc *alloc, slot_type *slot) {
+    std::allocator_traits<Alloc>::destroy(*alloc, slot);
+  }
+  static const Key &key(const value_type &x) { return x; }
+  static const Key &key(const slot_type *slot) { return *slot; }
+  static value_type &element(slot_type *slot) { return *slot; }
+  static const value_type &element(const slot_type *slot) { return *slot; }
+};
+
+// Helper functions to do a boolean comparison of two keys given a boolean
+// or three-way comparator.
+// SFINAE prevents implicit conversions to bool (such as from int).
+template <typename Result>
+constexpr bool compare_result_as_less_than(const Result r) {
+  if constexpr (std::is_signed_v<Result>) {
+    return r < 0;
+  } else {
+    return r;
+  }
+}
+// An adapter class that converts a lower-bound compare into an upper-bound
+// compare. Note: there is no need to make a version of this adapter specialized
+// for key-compare-to functors because the upper-bound (the first value greater
+// than the input) is never an exact match.
+template <typename Compare>
+struct upper_bound_adapter {
+  explicit upper_bound_adapter(const Compare &c) : comp(c) {}
+  template <typename K, typename LK>
+  bool operator()(const K &a, const LK &b) const {
+    // Returns true when a is not greater than b.
+    return !compare_result_as_less_than(comp(b, a));
+  }
+private:
+  const Compare& comp;
+};
+
+enum class MatchKind : uint8_t { kEq, kNe };
+
+template <typename V, bool IsCompareTo>
+struct SearchResult {
+  V value;
+  MatchKind match;
+
+  static constexpr bool has_match = true;
+  bool IsEq() const { return match == MatchKind::kEq; }
+};
+
+// When we don't use CompareTo, `match` is not present.
+// This ensures that callers can't use it accidentally when it provides no
+// useful information.
+template <typename V>
+struct SearchResult<V, false> {
+  V value;
+
+  static constexpr bool has_match = false;
+  static constexpr bool IsEq() { return false; }
+};
+
+// A node in the btree holding. The same node type is used for both internal
+// and leaf nodes in the btree, though the nodes are allocated in such a way
+// that the children array is only valid in internal nodes.
+template <typename Params>
+class btree_node {
+  using is_key_compare_to = typename Params::is_key_compare_to;
+  using is_multi_container = typename Params::is_multi_container;
+  using field_type = typename Params::node_count_type;
+  using allocator_type = typename Params::allocator_type;
+  using slot_type = typename Params::slot_type;
+
+ public:
+  using params_type = Params;
+  using key_type = typename Params::key_type;
+  using value_type = typename Params::value_type;
+  using mutable_value_type = typename Params::mutable_value_type;
+  using pointer = typename Params::pointer;
+  using const_pointer = typename Params::const_pointer;
+  using reference = typename Params::reference;
+  using const_reference = typename Params::const_reference;
+  using key_compare = typename Params::key_compare;
+  using size_type = typename Params::size_type;
+  using difference_type = typename Params::difference_type;
+
+  // Btree decides whether to use linear node search as follows:
+  //   - If the key is arithmetic and the comparator is std::less or
+  //     std::greater, choose linear.
+  //   - Otherwise, choose binary.
+  // TODO(ezb): Might make sense to add condition(s) based on node-size.
+  using use_linear_search = std::integral_constant<
+      bool,
+      std::is_arithmetic_v<key_type> &&
+      (std::is_same_v<std::less<key_type>, key_compare> ||
+       std::is_same_v<std::greater<key_type>, key_compare>)>;
+
+  ~btree_node() = default;
+  btree_node(const btree_node&) = delete;
+  btree_node& operator=(const btree_node&) = delete;
+
+ protected:
+  btree_node() = default;
+
+ private:
+  constexpr static size_type SizeWithNValues(size_type n) {
+    return sizeof(base_fields) + n * sizeof(value_type);;
+  }
+  // A lower bound for the overhead of fields other than values in a leaf node.
+  constexpr static size_type MinimumOverhead() {
+    return SizeWithNValues(1) - sizeof(value_type);
+  }
+
+  // Compute how many values we can fit onto a leaf node taking into account
+  // padding.
+  constexpr static size_type NodeTargetValues(const int begin, const int end) {
+    return begin == end ? begin
+                        : SizeWithNValues((begin + end) / 2 + 1) >
+                                  params_type::kTargetNodeSize
+                              ? NodeTargetValues(begin, (begin + end) / 2)
+                              : NodeTargetValues((begin + end) / 2 + 1, end);
+  }
+
+  constexpr static int kValueSize = params_type::kValueSize;
+  constexpr static int kTargetNodeSize = params_type::kTargetNodeSize;
+  constexpr static int kNodeTargetValues = NodeTargetValues(0, kTargetNodeSize);
+
+  // We need a minimum of 3 values per internal node in order to perform
+  // splitting (1 value for the two nodes involved in the split and 1 value
+  // propagated to the parent as the delimiter for the split).
+  constexpr static size_type kNodeValues = std::max(kNodeTargetValues, 3);
+
+  // The node is internal (i.e. is not a leaf node) if and only if `max_count`
+  // has this value.
+  constexpr static size_type kInternalNodeMaxCount = 0;
+
+  struct base_fields {
+    // A pointer to the node's parent.
+    btree_node *parent;
+    // The position of the node in the node's parent.
+    field_type position;
+    // The count of the number of values in the node.
+    field_type count;
+    // The maximum number of values the node can hold.
+    field_type max_count;
+  };
+
+  struct leaf_fields : public base_fields {
+    // The array of values. Only the first count of these values have been
+    // constructed and are valid.
+    slot_type values[kNodeValues];
+  };
+
+  struct internal_fields : public leaf_fields {
+    // The array of child pointers. The keys in children_[i] are all less than
+    // key(i). The keys in children_[i + 1] are all greater than key(i). There
+    // are always count + 1 children.
+    btree_node *children[kNodeValues + 1];
+  };
+
+  constexpr static size_type LeafSize(const int max_values = kNodeValues) {
+    return SizeWithNValues(max_values);
+  }
+  constexpr static size_type InternalSize() {
+    return sizeof(internal_fields);
+  }
+
+  template<auto MemPtr>
+  auto& GetField() {
+    return reinterpret_cast<internal_fields*>(this)->*MemPtr;
+  }
+
+  template<auto MemPtr>
+  auto& GetField() const {
+    return reinterpret_cast<const internal_fields*>(this)->*MemPtr;
+  }
+
+  void set_parent(btree_node *p) { GetField<&base_fields::parent>() = p; }
+  field_type &mutable_count() { return GetField<&base_fields::count>(); }
+  slot_type *slot(int i) { return &GetField<&leaf_fields::values>()[i]; }
+  const slot_type *slot(int i) const { return &GetField<&leaf_fields::values>()[i]; }
+  void set_position(field_type v) { GetField<&base_fields::position>() = v; }
+  void set_count(field_type v) { GetField<&base_fields::count>() = v; }
+  // This method is only called by the node init methods.
+  void set_max_count(field_type v) { GetField<&base_fields::max_count>() = v; }
+
+public:
+  constexpr static size_type Alignment() {
+    static_assert(alignof(leaf_fields) == alignof(internal_fields),
+                  "Alignment of all nodes must be equal.");
+    return alignof(internal_fields);
+  }
+
+  // Getter/setter for whether this is a leaf node or not. This value doesn't
+  // change after the node is created.
+  bool leaf() const { return GetField<&base_fields::max_count>() != kInternalNodeMaxCount; }
+
+  // Getter for the position of this node in its parent.
+  field_type position() const { return GetField<&base_fields::position>(); }
+
+  // Getter for the number of values stored in this node.
+  field_type count() const { return GetField<&base_fields::count>(); }
+  field_type max_count() const {
+    // Internal nodes have max_count==kInternalNodeMaxCount.
+    // Leaf nodes have max_count in [1, kNodeValues].
+    const field_type max_count = GetField<&base_fields::max_count>();
+    return max_count == field_type{kInternalNodeMaxCount}
+               ? field_type{kNodeValues}
+               : max_count;
+  }
+
+  // Getter for the parent of this node.
+  btree_node* parent() const { return GetField<&base_fields::parent>(); }
+  // Getter for whether the node is the root of the tree. The parent of the
+  // root of the tree is the leftmost node in the tree which is guaranteed to
+  // be a leaf.
+  bool is_root() const { return parent()->leaf(); }
+  void make_root() {
+    assert(parent()->is_root());
+    set_parent(parent()->parent());
+  }
+
+  // Getters for the key/value at position i in the node.
+  const key_type& key(int i) const { return params_type::key(slot(i)); }
+  reference value(int i) { return params_type::element(slot(i)); }
+  const_reference value(int i) const { return params_type::element(slot(i)); }
+
+  // Getters/setter for the child at position i in the node.
+  btree_node* child(int i) const { return GetField<&internal_fields::children>()[i]; }
+  btree_node*& mutable_child(int i) { return GetField<&internal_fields::children>()[i]; }
+  void clear_child(int i) {
+#ifndef NDEBUG
+    memset(&mutable_child(i), 0, sizeof(btree_node*));
+#endif
+  }
+  void set_child(int i, btree_node *c) {
+    mutable_child(i) = c;
+    c->set_position(i);
+  }
+  void init_child(int i, btree_node *c) {
+    set_child(i, c);
+    c->set_parent(this);
+  }
+  // Returns the position of the first value whose key is not less than k.
+  template <typename K>
+  SearchResult<int, is_key_compare_to::value> lower_bound(
+      const K &k, const key_compare &comp) const {
+    return use_linear_search::value ? linear_search(k, comp)
+                                    : binary_search(k, comp);
+  }
+  // Returns the position of the first value whose key is greater than k.
+  template <typename K>
+  int upper_bound(const K &k, const key_compare &comp) const {
+    auto upper_compare = upper_bound_adapter<key_compare>(comp);
+    return use_linear_search::value ? linear_search(k, upper_compare).value
+                                    : binary_search(k, upper_compare).value;
+  }
+
+  template <typename K, typename Compare>
+  SearchResult<int, btree_is_key_compare_to<Compare, key_type>::value>
+  linear_search(const K &k, const Compare &comp) const {
+    return linear_search_impl(k, 0, count(), comp,
+                              btree_is_key_compare_to<Compare, key_type>());
+  }
+
+  template <typename K, typename Compare>
+  SearchResult<int, btree_is_key_compare_to<Compare, key_type>::value>
+  binary_search(const K &k, const Compare &comp) const {
+    return binary_search_impl(k, 0, count(), comp,
+                              btree_is_key_compare_to<Compare, key_type>());
+  }
+  // Returns the position of the first value whose key is not less than k using
+  // linear search performed using plain compare.
+  template <typename K, typename Compare>
+  SearchResult<int, false> linear_search_impl(
+      const K &k, int s, const int e, const Compare &comp,
+      std::false_type /* IsCompareTo */) const {
+    while (s < e) {
+      if (!comp(key(s), k)) {
+        break;
+      }
+      ++s;
+    }
+    return {s};
+  }
+
+  // Returns the position of the first value whose key is not less than k using
+  // linear search performed using compare-to.
+  template <typename K, typename Compare>
+  SearchResult<int, true> linear_search_impl(
+      const K &k, int s, const int e, const Compare &comp,
+      std::true_type /* IsCompareTo */) const {
+    while (s < e) {
+      const auto c = comp(key(s), k);
+      if (c == 0) {
+        return {s, MatchKind::kEq};
+      } else if (c > 0) {
+        break;
+      }
+      ++s;
+    }
+    return {s, MatchKind::kNe};
+  }
+
+  // Returns the position of the first value whose key is not less than k using
+  // binary search performed using plain compare.
+  template <typename K, typename Compare>
+  SearchResult<int, false> binary_search_impl(
+      const K &k, int s, int e, const Compare &comp,
+      std::false_type /* IsCompareTo */) const {
+    while (s != e) {
+      const int mid = (s + e) >> 1;
+      if (comp(key(mid), k)) {
+        s = mid + 1;
+      } else {
+        e = mid;
+      }
+    }
+    return {s};
+  }
+
+  // Returns the position of the first value whose key is not less than k using
+  // binary search performed using compare-to.
+  template <typename K, typename CompareTo>
+  SearchResult<int, true> binary_search_impl(
+      const K &k, int s, int e, const CompareTo &comp,
+      std::true_type /* IsCompareTo */) const {
+    if constexpr (is_multi_container::value) {
+      MatchKind exact_match = MatchKind::kNe;
+      while (s != e) {
+        const int mid = (s + e) >> 1;
+        const auto c = comp(key(mid), k);
+        if (c < 0) {
+          s = mid + 1;
+        } else {
+          e = mid;
+          if (c == 0) {
+            // Need to return the first value whose key is not less than k,
+            // which requires continuing the binary search if this is a
+            // multi-container.
+            exact_match = MatchKind::kEq;
+          }
+        }
+      }
+      return {s, exact_match};
+    } else {  // Not a multi-container.
+      while (s != e) {
+        const int mid = (s + e) >> 1;
+        const auto c = comp(key(mid), k);
+        if (c < 0) {
+          s = mid + 1;
+        } else if (c > 0) {
+          e = mid;
+        } else {
+          return {mid, MatchKind::kEq};
+        }
+      }
+      return {s, MatchKind::kNe};
+    }
+  }
+
+  // Emplaces a value at position i, shifting all existing values and
+  // children at positions >= i to the right by 1.
+  template <typename... Args>
+  void emplace_value(size_type i, allocator_type *alloc, Args &&... args);
+
+  // Removes the value at position i, shifting all existing values and children
+  // at positions > i to the left by 1.
+  void remove_value(const int i, allocator_type *alloc);
+
+  // Removes the values at positions [i, i + to_erase), shifting all values
+  // after that range to the left by to_erase. Does not change children at all.
+  void remove_values_ignore_children(int i, int to_erase,
+                                     allocator_type *alloc);
+
+  // Rebalances a node with its right sibling.
+  void rebalance_right_to_left(const int to_move, btree_node *right,
+                               allocator_type *alloc);
+  void rebalance_left_to_right(const int to_move, btree_node *right,
+                               allocator_type *alloc);
+
+  // Splits a node, moving a portion of the node's values to its right sibling.
+  void split(const int insert_position, btree_node *dest, allocator_type *alloc);
+
+  // Merges a node with its right sibling, moving all of the values and the
+  // delimiting key in the parent node onto itself.
+  void merge(btree_node *sibling, allocator_type *alloc);
+
+  // Swap the contents of "this" and "src".
+  void swap(btree_node *src, allocator_type *alloc);
+
+  // Node allocation/deletion routines.
+  static btree_node *init_leaf(btree_node *n, btree_node *parent,
+                               int max_count) {
+    n->set_parent(parent);
+    n->set_position(0);
+    n->set_count(0);
+    n->set_max_count(max_count);
+    return n;
+  }
+  static btree_node *init_internal(btree_node *n, btree_node *parent) {
+    init_leaf(n, parent, kNodeValues);
+    // Set `max_count` to a sentinel value to indicate that this node is
+    // internal.
+    n->set_max_count(kInternalNodeMaxCount);
+    return n;
+  }
+  void destroy(allocator_type *alloc) {
+    for (int i = 0; i < count(); ++i) {
+      value_destroy(i, alloc);
+    }
+  }
+
+ private:
+  template <typename... Args>
+  void value_init(const size_type i, allocator_type *alloc, Args &&... args) {
+    params_type::construct(alloc, slot(i), std::forward<Args>(args)...);
+  }
+  void value_destroy(const size_type i, allocator_type *alloc) {
+    params_type::destroy(alloc, slot(i));
+  }
+
+  // Move n values starting at value i in this node into the values starting at
+  // value j in node x.
+  void uninitialized_move_n(const size_type n, const size_type i,
+                            const size_type j, btree_node *x,
+                            allocator_type *alloc) {
+    for (slot_type *src = slot(i), *end = src + n, *dest = x->slot(j);
+         src != end; ++src, ++dest) {
+      params_type::construct(alloc, dest, src);
+    }
+  }
+
+  // Destroys a range of n values, starting at index i.
+  void value_destroy_n(const size_type i, const size_type n,
+                       allocator_type *alloc) {
+    for (int j = 0; j < n; ++j) {
+      value_destroy(i + j, alloc);
+    }
+  }
+
+private:
+  template <typename P>
+  friend class btree;
+  template <typename N, typename R, typename P>
+  friend struct btree_iterator;
+};
+
+template <typename Node, typename Reference, typename Pointer>
+struct btree_iterator {
+ private:
+  using key_type = typename Node::key_type;
+  using size_type = typename Node::size_type;
+  using params_type = typename Node::params_type;
+
+  using node_type = Node;
+  using normal_node = typename std::remove_const<Node>::type;
+  using const_node = const Node;
+  using normal_pointer = typename params_type::pointer;
+  using normal_reference = typename params_type::reference;
+  using const_pointer = typename params_type::const_pointer;
+  using const_reference = typename params_type::const_reference;
+  using slot_type = typename params_type::slot_type;
+
+  using iterator =
+      btree_iterator<normal_node, normal_reference, normal_pointer>;
+  using const_iterator =
+      btree_iterator<const_node, const_reference, const_pointer>;
+
+ public:
+  // These aliases are public for std::iterator_traits.
+  using difference_type = typename Node::difference_type;
+  using value_type = typename params_type::value_type;
+  using pointer = Pointer;
+  using reference = Reference;
+  using iterator_category = std::bidirectional_iterator_tag;
+
+  btree_iterator() = default;
+  btree_iterator(Node *n, int p) : node(n), position(p) {}
+
+  // NOTE: this SFINAE allows for implicit conversions from iterator to
+  // const_iterator, but it specifically avoids defining copy constructors so
+  // that btree_iterator can be trivially copyable. This is for performance and
+  // binary size reasons.
+  template<typename N, typename R, typename P,
+           std::enable_if_t<
+             std::is_same_v<btree_iterator<N, R, P>, iterator> &&
+             std::is_same_v<btree_iterator, const_iterator>,
+             int> = 0>
+  btree_iterator(const btree_iterator<N, R, P> &x)
+      : node(x.node), position(x.position) {}
+
+ private:
+  // This SFINAE allows explicit conversions from const_iterator to
+  // iterator, but also avoids defining a copy constructor.
+  // NOTE: the const_cast is safe because this constructor is only called by
+  // non-const methods and the container owns the nodes.
+  template <typename N, typename R, typename P,
+            std::enable_if_t<
+              std::is_same_v<btree_iterator<N, R, P>, const_iterator> &&
+              std::is_same_v<btree_iterator, iterator>,
+              int> = 0>
+  explicit btree_iterator(const btree_iterator<N, R, P> &x)
+      : node(const_cast<node_type *>(x.node)), position(x.position) {}
+
+  // Increment/decrement the iterator.
+  void increment() {
+    if (node->leaf() && ++position < node->count()) {
+      return;
+    }
+    increment_slow();
+  }
+  void increment_slow();
+
+  void decrement() {
+    if (node->leaf() && --position >= 0) {
+      return;
+    }
+    decrement_slow();
+  }
+  void decrement_slow();
+
+ public:
+  bool operator==(const const_iterator &x) const {
+    return node == x.node && position == x.position;
+  }
+  bool operator!=(const const_iterator &x) const {
+    return node != x.node || position != x.position;
+  }
+
+  // Accessors for the key/value the iterator is pointing at.
+  reference operator*() const {
+    return node->value(position);
+  }
+  pointer operator->() const {
+    return &node->value(position);
+  }
+
+  btree_iterator& operator++() {
+    increment();
+    return *this;
+  }
+  btree_iterator& operator--() {
+    decrement();
+    return *this;
+  }
+  btree_iterator operator++(int) {
+    btree_iterator tmp = *this;
+    ++*this;
+    return tmp;
+  }
+  btree_iterator operator--(int) {
+    btree_iterator tmp = *this;
+    --*this;
+    return tmp;
+  }
+
+ private:
+  template <typename Params>
+  friend class btree;
+  template <typename Tree>
+  friend class btree_container;
+  template <typename Tree>
+  friend class btree_set_container;
+  template <typename Tree>
+  friend class btree_map_container;
+  template <typename Tree>
+  friend class btree_multiset_container;
+  template <typename N, typename R, typename P>
+  friend struct btree_iterator;
+
+  const key_type &key() const { return node->key(position); }
+  slot_type *slot() { return node->slot(position); }
+
+  // The node in the tree the iterator is pointing at.
+  Node *node = nullptr;
+  // The position within the node of the tree the iterator is pointing at.
+  int position = -1;
+};
+
+template <size_t Alignment, class Alloc>
+class AlignedAlloc {
+  struct alignas(Alignment) M {};
+  using alloc_t =
+    typename std::allocator_traits<Alloc>::template rebind_alloc<M>;
+  using traits_t =
+    typename std::allocator_traits<Alloc>::template rebind_traits<M>;
+  static constexpr size_t num_aligned_objects(size_t size) {
+    return (size + sizeof(M) - 1) / sizeof(M);
+  }
+public:
+  static void* allocate(Alloc* alloc, size_t size) {
+    alloc_t aligned_alloc(*alloc);
+    void* p = traits_t::allocate(aligned_alloc,
+                                 num_aligned_objects(size));
+    assert(reinterpret_cast<uintptr_t>(p) % Alignment == 0 &&
+         "allocator does not respect alignment");
+    return p;
+  }
+  static void deallocate(Alloc* alloc, void* p, size_t size) {
+    alloc_t aligned_alloc(*alloc);
+    traits_t::deallocate(aligned_alloc, static_cast<M*>(p),
+                         num_aligned_objects(size));
+  }
+};
+
+template <typename Params>
+class btree {
+  using node_type = btree_node<Params>;
+  using is_key_compare_to = typename Params::is_key_compare_to;
+
+  // We use a static empty node for the root/leftmost/rightmost of empty btrees
+  // in order to avoid branching in begin()/end().
+  struct alignas(node_type::Alignment()) EmptyNodeType : node_type {
+    using field_type = typename node_type::field_type;
+    node_type *parent;
+    field_type position = 0;
+    field_type count = 0;
+    // max_count must be != kInternalNodeMaxCount (so that this node is regarded
+    // as a leaf node). max_count() is never called when the tree is empty.
+    field_type max_count = node_type::kInternalNodeMaxCount + 1;
+
+    constexpr EmptyNodeType(node_type *p) : parent(p) {}
+  };
+
+  static node_type *EmptyNode() {
+    static constexpr EmptyNodeType empty_node(
+        const_cast<EmptyNodeType *>(&empty_node));
+    return const_cast<EmptyNodeType *>(&empty_node);
+  }
+
+  constexpr static int kNodeValues = node_type::kNodeValues;
+  constexpr static int kMinNodeValues = kNodeValues / 2;
+  constexpr static int kValueSize = node_type::kValueSize;
+
+  // A helper class to get the empty base class optimization for 0-size
+  // allocators. Base is allocator_type.
+  // (e.g. empty_base_handle<key_compare, allocator_type, node_type*>). If Base is
+  // 0-size, the compiler doesn't have to reserve any space for it and
+  // sizeof(empty_base_handle) will simply be sizeof(Data). Google [empty base
+  // class optimization] for more details.
+  template <typename Base1, typename Base2, typename Data>
+  struct empty_base_handle : public Base1, Base2 {
+    empty_base_handle(const Base1 &b1, const Base2 &b2, const Data &d)
+        : Base1(b1),
+          Base2(b2),
+          data(d) {}
+    Data data;
+  };
+
+  struct node_stats {
+    using size_type = typename Params::size_type;
+
+    node_stats(size_type l, size_type i)
+        : leaf_nodes(l),
+          internal_nodes(i) {
+    }
+
+    node_stats& operator+=(const node_stats &x) {
+      leaf_nodes += x.leaf_nodes;
+      internal_nodes += x.internal_nodes;
+      return *this;
+    }
+
+    size_type leaf_nodes;
+    size_type internal_nodes;
+  };
+
+ public:
+  using key_type = typename Params::key_type;
+  using value_type = typename Params::value_type;
+  using size_type = typename Params::size_type;
+  using difference_type = typename Params::difference_type;
+  using key_compare = typename Params::key_compare;
+  using value_compare = typename Params::value_compare;
+  using allocator_type = typename Params::allocator_type;
+  using reference = typename Params::reference;
+  using const_reference = typename Params::const_reference;
+  using pointer = typename Params::pointer;
+  using const_pointer = typename Params::const_pointer;
+  using iterator = btree_iterator<node_type, reference, pointer>;
+  using const_iterator = typename iterator::const_iterator;
+  using reverse_iterator = std::reverse_iterator<iterator>;
+  using const_reverse_iterator = std::reverse_iterator<const_iterator>;
+
+  // Internal types made public for use by btree_container types.
+  using params_type = Params;
+
+ private:
+  // For use in copy_or_move_values_in_order.
+  const value_type &maybe_move_from_iterator(const_iterator x) { return *x; }
+  value_type &&maybe_move_from_iterator(iterator x) { return std::move(*x); }
+
+  // Copies or moves (depending on the template parameter) the values in
+  // x into this btree in their order in x. This btree must be empty before this
+  // method is called. This method is used in copy construction, copy
+  // assignment, and move assignment.
+  template <typename Btree>
+  void copy_or_move_values_in_order(Btree *x);
+
+  // Validates that various assumptions/requirements are true at compile time.
+  constexpr static bool static_assert_validation();
+
+ public:
+  btree(const key_compare &comp, const allocator_type &alloc);
+
+  btree(const btree &x);
+  btree(btree &&x) noexcept
+      : root_(std::move(x.root_)),
+        rightmost_(std::exchange(x.rightmost_, EmptyNode())),
+        size_(std::exchange(x.size_, 0)) {
+    x.mutable_root() = EmptyNode();
+  }
+
+  ~btree() {
+    // Put static_asserts in destructor to avoid triggering them before the type
+    // is complete.
+    static_assert(static_assert_validation(), "This call must be elided.");
+    clear();
+  }
+
+  // Assign the contents of x to *this.
+  btree &operator=(const btree &x);
+  btree &operator=(btree &&x) noexcept;
+
+  iterator begin() {
+    return iterator(leftmost(), 0);
+  }
+  const_iterator begin() const {
+    return const_iterator(leftmost(), 0);
+  }
+  iterator end() {
+    return iterator(rightmost_, rightmost_->count());
+  }
+  const_iterator end() const {
+    return const_iterator(rightmost_, rightmost_->count());
+  }
+  reverse_iterator rbegin() {
+    return reverse_iterator(end());
+  }
+  const_reverse_iterator rbegin() const {
+    return const_reverse_iterator(end());
+  }
+  reverse_iterator rend() {
+    return reverse_iterator(begin());
+  }
+  const_reverse_iterator rend() const {
+    return const_reverse_iterator(begin());
+  }
+
+  // Finds the first element whose key is not less than key.
+  template <typename K>
+  iterator lower_bound(const K &key) {
+    return internal_end(internal_lower_bound(key));
+  }
+  template <typename K>
+  const_iterator lower_bound(const K &key) const {
+    return internal_end(internal_lower_bound(key));
+  }
+
+  // Finds the first element whose key is greater than key.
+  template <typename K>
+  iterator upper_bound(const K &key) {
+    return internal_end(internal_upper_bound(key));
+  }
+  template <typename K>
+  const_iterator upper_bound(const K &key) const {
+    return internal_end(internal_upper_bound(key));
+  }
+
+  // Finds the range of values which compare equal to key. The first member of
+  // the returned pair is equal to lower_bound(key). The second member pair of
+  // the pair is equal to upper_bound(key).
+  template <typename K>
+  std::pair<iterator, iterator> equal_range(const K &key) {
+    return {lower_bound(key), upper_bound(key)};
+  }
+  template <typename K>
+  std::pair<const_iterator, const_iterator> equal_range(const K &key) const {
+    return {lower_bound(key), upper_bound(key)};
+  }
+
+  // Inserts a value into the btree only if it does not already exist. The
+  // boolean return value indicates whether insertion succeeded or failed.
+  // Requirement: if `key` already exists in the btree, does not consume `args`.
+  // Requirement: `key` is never referenced after consuming `args`.
+  template <typename... Args>
+  std::pair<iterator, bool> insert_unique(const key_type &key, Args &&... args);
+
+  // Inserts with hint. Checks to see if the value should be placed immediately
+  // before `position` in the tree. If so, then the insertion will take
+  // amortized constant time. If not, the insertion will take amortized
+  // logarithmic time as if a call to insert_unique() were made.
+  // Requirement: if `key` already exists in the btree, does not consume `args`.
+  // Requirement: `key` is never referenced after consuming `args`.
+  template <typename... Args>
+  std::pair<iterator, bool> insert_hint_unique(iterator position,
+                                               const key_type &key,
+                                               Args &&... args);
+
+  // Insert a range of values into the btree.
+  template <typename InputIterator>
+  void insert_iterator_unique(InputIterator b, InputIterator e);
+
+  // Inserts a value into the btree.
+  template <typename ValueType>
+  iterator insert_multi(const key_type &key, ValueType &&v);
+
+  // Inserts a value into the btree.
+  template <typename ValueType>
+  iterator insert_multi(ValueType &&v) {
+    return insert_multi(params_type::key(v), std::forward<ValueType>(v));
+  }
+
+  // Insert with hint. Check to see if the value should be placed immediately
+  // before position in the tree. If it does, then the insertion will take
+  // amortized constant time. If not, the insertion will take amortized
+  // logarithmic time as if a call to insert_multi(v) were made.
+  template <typename ValueType>
+  iterator insert_hint_multi(iterator position, ValueType &&v);
+
+  // Insert a range of values into the btree.
+  template <typename InputIterator>
+  void insert_iterator_multi(InputIterator b, InputIterator e);
+
+  // Erase the specified iterator from the btree. The iterator must be valid
+  // (i.e. not equal to end()).  Return an iterator pointing to the node after
+  // the one that was erased (or end() if none exists).
+  // Requirement: does not read the value at `*iter`.
+  iterator erase(iterator iter);
+
+  // Erases range. Returns the number of keys erased and an iterator pointing
+  // to the element after the last erased element.
+  std::pair<size_type, iterator> erase(iterator begin, iterator end);
+
+  // Erases the specified key from the btree. Returns 1 if an element was
+  // erased and 0 otherwise.
+  template <typename K>
+  size_type erase_unique(const K &key);
+
+  // Erases all of the entries matching the specified key from the
+  // btree. Returns the number of elements erased.
+  template <typename K>
+  size_type erase_multi(const K &key);
+
+  // Finds the iterator corresponding to a key or returns end() if the key is
+  // not present.
+  template <typename K>
+  iterator find(const K &key) {
+    return internal_end(internal_find(key));
+  }
+  template <typename K>
+  const_iterator find(const K &key) const {
+    return internal_end(internal_find(key));
+  }
+
+  // Returns a count of the number of times the key appears in the btree.
+  template <typename K>
+  size_type count_unique(const K &key) const {
+    const iterator begin = internal_find(key);
+    if (begin.node == nullptr) {
+      // The key doesn't exist in the tree.
+      return 0;
+    }
+    return 1;
+  }
+  // Returns a count of the number of times the key appears in the btree.
+  template <typename K>
+  size_type count_multi(const K &key) const {
+    const auto range = equal_range(key);
+    return std::distance(range.first, range.second);
+  }
+
+  // Clear the btree, deleting all of the values it contains.
+  void clear();
+
+  // Swap the contents of *this and x.
+  void swap(btree &x);
+
+  const key_compare &key_comp() const noexcept {
+    return *static_cast<const key_compare*>(&root_);
+  }
+  template <typename K, typename LK>
+  bool compare_keys(const K &x, const LK &y) const {
+    return compare_result_as_less_than(key_comp()(x, y));
+  }
+
+  // Verifies the structure of the btree.
+  void verify() const;
+
+  // Size routines.
+  size_type size() const { return size_; }
+  size_type max_size() const { return std::numeric_limits<size_type>::max(); }
+  bool empty() const { return size_ == 0; }
+
+  // The height of the btree. An empty tree will have height 0.
+  size_type height() const {
+    size_type h = 0;
+    if (!empty()) {
+      // Count the length of the chain from the leftmost node up to the
+      // root. We actually count from the root back around to the level below
+      // the root, but the calculation is the same because of the circularity
+      // of that traversal.
+      const node_type *n = root();
+      do {
+        ++h;
+        n = n->parent();
+      } while (n != root());
+    }
+    return h;
+  }
+
+  // The number of internal, leaf and total nodes used by the btree.
+  size_type leaf_nodes() const {
+    return internal_stats(root()).leaf_nodes;
+  }
+  size_type internal_nodes() const {
+    return internal_stats(root()).internal_nodes;
+  }
+  size_type nodes() const {
+    node_stats stats = internal_stats(root());
+    return stats.leaf_nodes + stats.internal_nodes;
+  }
+
+  // The total number of bytes used by the btree.
+  size_type bytes_used() const {
+    node_stats stats = internal_stats(root());
+    if (stats.leaf_nodes == 1 && stats.internal_nodes == 0) {
+      return sizeof(*this) +
+             node_type::LeafSize(root()->max_count());
+    } else {
+      return sizeof(*this) +
+             stats.leaf_nodes * node_type::LeafSize() +
+             stats.internal_nodes * node_type::InternalSize();
+    }
+  }
+
+  // The average number of bytes used per value stored in the btree.
+  static double average_bytes_per_value() {
+    // Returns the number of bytes per value on a leaf node that is 75%
+    // full. Experimentally, this matches up nicely with the computed number of
+    // bytes per value in trees that had their values inserted in random order.
+    return node_type::LeafSize() / (kNodeValues * 0.75);
+  }
+
+  // The fullness of the btree. Computed as the number of elements in the btree
+  // divided by the maximum number of elements a tree with the current number
+  // of nodes could hold. A value of 1 indicates perfect space
+  // utilization. Smaller values indicate space wastage.
+  // Returns 0 for empty trees.
+  double fullness() const {
+    if (empty()) return 0.0;
+    return static_cast<double>(size()) / (nodes() * kNodeValues);
+  }
+  // The overhead of the btree structure in bytes per node. Computed as the
+  // total number of bytes used by the btree minus the number of bytes used for
+  // storing elements divided by the number of elements.
+  // Returns 0 for empty trees.
+  double overhead() const {
+    if (empty()) return 0.0;
+    return (bytes_used() - size() * sizeof(value_type)) /
+           static_cast<double>(size());
+  }
+
+  // The allocator used by the btree.
+  allocator_type get_allocator() const {
+    return allocator();
+  }
+
+ private:
+  // Internal accessor routines.
+  node_type *root() { return root_.data; }
+  const node_type *root() const { return root_.data; }
+  node_type *&mutable_root() { return root_.data; }
+  key_compare *mutable_key_comp() noexcept {
+    return static_cast<key_compare*>(&root_);
+  }
+
+  node_type* rightmost() {
+    return rightmost_;
+  }
+  const node_type* rightmost() const {
+    return rightmost_;
+  }
+  // The leftmost node is stored as the parent of the root node.
+  node_type* leftmost() { return root() ? root()->parent() : NULL; }
+  const node_type* leftmost() const { return root() ? root()->parent() : NULL; }
+
+  // The size of the tree is stored in the root node.
+  size_type* mutable_size() { return root()->mutable_size(); }
+
+  // Allocator routines.
+  allocator_type* mutable_allocator() noexcept {
+    return static_cast<allocator_type*>(&root_);
+  }
+  const allocator_type& allocator() const noexcept {
+    return *static_cast<const allocator_type*>(&root_);
+  }
+
+  node_type *allocate(const size_type size) {
+    using aligned_alloc_t =
+      AlignedAlloc<node_type::Alignment(), allocator_type>;
+    return static_cast<node_type*>(
+      aligned_alloc_t::allocate(mutable_allocator(), size));
+  }
+
+  // Node creation/deletion routines.
+  node_type* new_internal_node(node_type *parent) {
+    node_type *p = allocate(node_type::InternalSize());
+    return node_type::init_internal(p, parent);
+  }
+  node_type* new_leaf_node(node_type *parent) {
+    node_type *p = allocate(node_type::LeafSize());
+    return node_type::init_leaf(p, parent, kNodeValues);
+  }
+  node_type *new_leaf_root_node(const int max_count) {
+    node_type *p = allocate(node_type::LeafSize(max_count));
+    return node_type::init_leaf(p, p, max_count);
+  }
+
+  // Deletion helper routines.
+  void erase_same_node(iterator begin, iterator end);
+  iterator erase_from_leaf_node(iterator begin, size_type to_erase);
+  iterator rebalance_after_delete(iterator iter);
+
+  // Deallocates a node of a certain size in bytes using the allocator.
+  void deallocate(const size_type size, node_type *node) {
+    using aligned_alloc_t =
+      AlignedAlloc<node_type::Alignment(), allocator_type>;
+    aligned_alloc_t::deallocate(mutable_allocator(), node, size);
+  }
+
+  void delete_internal_node(node_type *node) {
+    node->destroy(mutable_allocator());
+    deallocate(node_type::InternalSize(), node);
+  }
+  void delete_leaf_node(node_type *node) {
+    node->destroy(mutable_allocator());
+    deallocate(node_type::LeafSize(node->max_count()), node);
+  }
+
+  // Rebalances or splits the node iter points to.
+  void rebalance_or_split(iterator *iter);
+
+  // Merges the values of left, right and the delimiting key on their parent
+  // onto left, removing the delimiting key and deleting right.
+  void merge_nodes(node_type *left, node_type *right);
+
+  // Tries to merge node with its left or right sibling, and failing that,
+  // rebalance with its left or right sibling. Returns true if a merge
+  // occurred, at which point it is no longer valid to access node. Returns
+  // false if no merging took place.
+  bool try_merge_or_rebalance(iterator *iter);
+
+  // Tries to shrink the height of the tree by 1.
+  void try_shrink();
+
+  iterator internal_end(iterator iter) {
+    return iter.node != nullptr ? iter : end();
+  }
+  const_iterator internal_end(const_iterator iter) const {
+    return iter.node != nullptr ? iter : end();
+  }
+
+  // Emplaces a value into the btree immediately before iter. Requires that
+  // key(v) <= iter.key() and (--iter).key() <= key(v).
+  template <typename... Args>
+  iterator internal_emplace(iterator iter, Args &&... args);
+
+  // Returns an iterator pointing to the first value >= the value "iter" is
+  // pointing at. Note that "iter" might be pointing to an invalid location as
+  // iter.position == iter.node->count(). This routine simply moves iter up in
+  // the tree to a valid location.
+  // Requires: iter.node is non-null.
+  template <typename IterType>
+  static IterType internal_last(IterType iter);
+
+  // Returns an iterator pointing to the leaf position at which key would
+  // reside in the tree. We provide 2 versions of internal_locate. The first
+  // version uses a less-than comparator and is incapable of distinguishing when
+  // there is an exact match. The second version is for the key-compare-to
+  // specialization and distinguishes exact matches. The key-compare-to
+  // specialization allows the caller to avoid a subsequent comparison to
+  // determine if an exact match was made, which is important for keys with
+  // expensive comparison, such as strings.
+  template <typename K>
+  SearchResult<iterator, is_key_compare_to::value> internal_locate(
+      const K &key) const;
+
+  template <typename K>
+  SearchResult<iterator, false> internal_locate_impl(
+      const K &key, std::false_type /* IsCompareTo */) const;
+
+  template <typename K>
+  SearchResult<iterator, true> internal_locate_impl(
+      const K &key, std::true_type /* IsCompareTo */) const;
+
+  // Internal routine which implements lower_bound().
+  template <typename K>
+  iterator internal_lower_bound(const K &key) const;
+
+  // Internal routine which implements upper_bound().
+  template <typename K>
+  iterator internal_upper_bound(const K &key) const;
+
+  // Internal routine which implements find().
+  template <typename K>
+  iterator internal_find(const K &key) const;
+
+  // Deletes a node and all of its children.
+  void internal_clear(node_type *node);
+
+  // Verifies the tree structure of node.
+  int internal_verify(const node_type *node,
+                      const key_type *lo, const key_type *hi) const;
+
+  node_stats internal_stats(const node_type *node) const {
+    // The root can be a static empty node.
+    if (node == nullptr || (node == root() && empty())) {
+      return node_stats(0, 0);
+    }
+    if (node->leaf()) {
+      return node_stats(1, 0);
+    }
+    node_stats res(0, 1);
+    for (int i = 0; i <= node->count(); ++i) {
+      res += internal_stats(node->child(i));
+    }
+    return res;
+  }
+
+ private:
+  empty_base_handle<key_compare, allocator_type, node_type*> root_;
+
+  // A pointer to the rightmost node. Note that the leftmost node is stored as
+  // the root's parent.
+  node_type *rightmost_;
+
+  // Number of values.
+  size_type size_;
+};
+
+////
+// btree_node methods
+template <typename P>
+template <typename... Args>
+inline void btree_node<P>::emplace_value(const size_type i,
+                                         allocator_type *alloc,
+                                         Args &&... args) {
+  assert(i <= count());
+  // Shift old values to create space for new value and then construct it in
+  // place.
+  if (i < count()) {
+    value_init(count(), alloc, slot(count() - 1));
+    std::copy_backward(std::make_move_iterator(slot(i)),
+                       std::make_move_iterator(slot(count() - 1)),
+                       slot(count()));
+    value_destroy(i, alloc);
+  }
+  value_init(i, alloc, std::forward<Args>(args)...);
+  set_count(count() + 1);
+
+  if (!leaf() && count() > i + 1) {
+    for (int j = count(); j > i + 1; --j) {
+      set_child(j, child(j - 1));
+    }
+    clear_child(i + 1);
+  }
+}
+
+template <typename P>
+inline void btree_node<P>::remove_value(const int i, allocator_type *alloc) {
+  if (!leaf() && count() > i + 1) {
+    assert(child(i + 1)->count() == 0);
+    for (size_type j = i + 1; j < count(); ++j) {
+      set_child(j, child(j + 1));
+    }
+    clear_child(count());
+  }
+
+  remove_values_ignore_children(i, /*to_erase=*/1, alloc);
+}
+
+template <typename P>
+inline void btree_node<P>::remove_values_ignore_children(
+    const int i, const int to_erase, allocator_type *alloc) {
+  assert(to_erase >= 0);
+  std::copy(std::make_move_iterator(slot(i + to_erase)),
+            std::make_move_iterator(slot(count())),
+            slot(i));
+  value_destroy_n(count() - to_erase, to_erase, alloc);
+  set_count(count() - to_erase);
+}
+
+template <typename P>
+void btree_node<P>::rebalance_right_to_left(const int to_move,
+                                            btree_node *right,
+                                            allocator_type *alloc) {
+  assert(parent() == right->parent());
+  assert(position() + 1 == right->position());
+  assert(right->count() >= count());
+  assert(to_move >= 1);
+  assert(to_move <= right->count());
+
+  // 1) Move the delimiting value in the parent to the left node.
+  value_init(count(), alloc, parent()->slot(position()));
+
+  // 2) Move the (to_move - 1) values from the right node to the left node.
+  right->uninitialized_move_n(to_move - 1, 0, count() + 1, this, alloc);
+
+  // 3) Move the new delimiting value to the parent from the right node.
+  params_type::move(alloc, right->slot(to_move - 1),
+                    parent()->slot(position()));
+
+  // 4) Shift the values in the right node to their correct position.
+  std::copy(std::make_move_iterator(right->slot(to_move)),
+            std::make_move_iterator(right->slot(right->count())),
+            right->slot(0));
+
+  // 5) Destroy the now-empty to_move entries in the right node.
+  right->value_destroy_n(right->count() - to_move, to_move, alloc);
+
+  if (!leaf()) {
+    // Move the child pointers from the right to the left node.
+    for (int i = 0; i < to_move; ++i) {
+      init_child(count() + i + 1, right->child(i));
+    }
+    for (int i = 0; i <= right->count() - to_move; ++i) {
+      assert(i + to_move <= right->max_count());
+      right->init_child(i, right->child(i + to_move));
+      right->clear_child(i + to_move);
+    }
+  }
+
+  // Fixup the counts on the left and right nodes.
+  set_count(count() + to_move);
+  right->set_count(right->count() - to_move);
+}
+
+template <typename P>
+void btree_node<P>::rebalance_left_to_right(const int to_move,
+                                            btree_node *right,
+                                            allocator_type *alloc) {
+  assert(parent() == right->parent());
+  assert(position() + 1 == right->position());
+  assert(count() >= right->count());
+  assert(to_move >= 1);
+  assert(to_move <= count());
+
+  // Values in the right node are shifted to the right to make room for the
+  // new to_move values. Then, the delimiting value in the parent and the
+  // other (to_move - 1) values in the left node are moved into the right node.
+  // Lastly, a new delimiting value is moved from the left node into the
+  // parent, and the remaining empty left node entries are destroyed.
+
+  if (right->count() >= to_move) {
+    // The original location of the right->count() values are sufficient to hold
+    // the new to_move entries from the parent and left node.
+
+    // 1) Shift existing values in the right node to their correct positions.
+    right->uninitialized_move_n(to_move, right->count() - to_move,
+                                right->count(), right, alloc);
+    std::copy_backward(std::make_move_iterator(right->slot(0)),
+                       std::make_move_iterator(right->slot(right->count() - to_move)),
+                       right->slot(right->count()));
+
+    // 2) Move the delimiting value in the parent to the right node.
+    params_type::move(alloc, parent()->slot(position()),
+                      right->slot(to_move - 1));
+
+    // 3) Move the (to_move - 1) values from the left node to the right node.
+    std::copy(std::make_move_iterator(slot(count() - (to_move - 1))),
+              std::make_move_iterator(slot(count())),
+              right->slot(0));
+  } else {
+    // The right node does not have enough initialized space to hold the new
+    // to_move entries, so part of them will move to uninitialized space.
+
+    // 1) Shift existing values in the right node to their correct positions.
+    right->uninitialized_move_n(right->count(), 0, to_move, right, alloc);
+
+    // 2) Move the delimiting value in the parent to the right node.
+    right->value_init(to_move - 1, alloc, parent()->slot(position()));
+
+    // 3) Move the (to_move - 1) values from the left node to the right node.
+    const size_type uninitialized_remaining = to_move - right->count() - 1;
+    uninitialized_move_n(uninitialized_remaining,
+                         count() - uninitialized_remaining, right->count(),
+                         right, alloc);
+    std::copy(std::make_move_iterator(slot(count() - (to_move - 1))),
+              std::make_move_iterator(slot(count() - uninitialized_remaining)),
+              right->slot(0));
+  }
+
+  // 4) Move the new delimiting value to the parent from the left node.
+  params_type::move(alloc, slot(count() - to_move), parent()->slot(position()));
+
+  // 5) Destroy the now-empty to_move entries in the left node.
+  value_destroy_n(count() - to_move, to_move, alloc);
+
+  if (!leaf()) {
+    // Move the child pointers from the left to the right node.
+    for (int i = right->count(); i >= 0; --i) {
+      right->init_child(i + to_move, right->child(i));
+      right->clear_child(i);
+    }
+    for (int i = 1; i <= to_move; ++i) {
+      right->init_child(i - 1, child(count() - to_move + i));
+      clear_child(count() - to_move + i);
+    }
+  }
+
+  // Fixup the counts on the left and right nodes.
+  set_count(count() - to_move);
+  right->set_count(right->count() + to_move);
+}
+
+template <typename P>
+void btree_node<P>::split(const int insert_position, btree_node *dest,
+                          allocator_type *alloc) {
+  assert(dest->count() == 0);
+  assert(max_count() == kNodeValues);
+
+  // We bias the split based on the position being inserted. If we're
+  // inserting at the beginning of the left node then bias the split to put
+  // more values on the right node. If we're inserting at the end of the
+  // right node then bias the split to put more values on the left node.
+  if (insert_position == 0) {
+    dest->set_count(count() - 1);
+  } else if (insert_position == kNodeValues) {
+    dest->set_count(0);
+  } else {
+    dest->set_count(count() / 2);
+  }
+  set_count(count() - dest->count());
+  assert(count() >= 1);
+
+  // Move values from the left sibling to the right sibling.
+  uninitialized_move_n(dest->count(), count(), 0, dest, alloc);
+
+  // Destroy the now-empty entries in the left node.
+  value_destroy_n(count(), dest->count(), alloc);
+
+  // The split key is the largest value in the left sibling.
+  set_count(count() - 1);
+  parent()->emplace_value(position(), alloc, slot(count()));
+  value_destroy(count(), alloc);
+  parent()->init_child(position() + 1, dest);
+
+  if (!leaf()) {
+    for (int i = 0; i <= dest->count(); ++i) {
+      assert(child(count() + i + 1) != nullptr);
+      dest->init_child(i, child(count() + i + 1));
+      clear_child(count() + i + 1);
+    }
+  }
+}
+
+template <typename P>
+void btree_node<P>::merge(btree_node *src, allocator_type *alloc) {
+  assert(parent() == src->parent());
+  assert(position() + 1 == src->position());
+
+  // Move the delimiting value to the left node.
+  value_init(count(), alloc, parent()->slot(position()));
+
+    // Move the values from the right to the left node.
+  src->uninitialized_move_n(src->count(), 0, count() + 1, this, alloc);
+
+  // Destroy the now-empty entries in the right node.
+  src->value_destroy_n(0, src->count(), alloc);
+
+  if (!leaf()) {
+    // Move the child pointers from the right to the left node.
+    for (int i = 0; i <= src->count(); ++i) {
+      init_child(count() + i + 1, src->child(i));
+      src->clear_child(i);
+    }
+  }
+
+  // Fixup the counts on the src and dest nodes.
+  set_count(1 + count() + src->count());
+  src->set_count(0);
+
+  // Remove the value on the parent node.
+  parent()->remove_value(position(), alloc);
+}
+
+template <typename P>
+void btree_node<P>::swap(btree_node *x, allocator_type *alloc) {
+  using std::swap;
+  assert(leaf() == x->leaf());
+
+  // Determine which is the smaller/larger node.
+  btree_node *smaller = this, *larger = x;
+  if (smaller->count() > larger->count()) {
+    swap(smaller, larger);
+  }
+
+  // Swap the values.
+  std::swap_ranges(smaller->slot(0), smaller->slot(smaller->count()),
+                   larger->slot(0));
+
+  // Move values that can't be swapped.
+  const size_type to_move = larger->count() - smaller->count();
+  larger->uninitialized_move_n(to_move, smaller->count(), smaller->count(),
+                               smaller, alloc);
+  larger->value_destroy_n(smaller->count(), to_move, alloc);
+
+  if (!leaf()) {
+    // Swap the child pointers.
+    std::swap_ranges(&smaller->mutable_child(0),
+                     &smaller->mutable_child(smaller->count() + 1),
+                     &larger->mutable_child(0));
+    // Update swapped children's parent pointers.
+    int i = 0;
+    for (; i <= smaller->count(); ++i) {
+      smaller->child(i)->set_parent(smaller);
+      larger->child(i)->set_parent(larger);
+    }
+    // Move the child pointers that couldn't be swapped.
+    for (; i <= larger->count(); ++i) {
+      smaller->init_child(i, larger->child(i));
+      larger->clear_child(i);
+    }
+  }
+
+  // Swap the counts.
+  swap(mutable_count(), x->mutable_count());
+}
+
+////
+// btree_iterator methods
+template <typename N, typename R, typename P>
+void btree_iterator<N, R, P>::increment_slow() {
+  if (node->leaf()) {
+    assert(position >= node->count());
+    btree_iterator save(*this);
+    while (position == node->count() && !node->is_root()) {
+      assert(node->parent()->child(node->position()) == node);
+      position = node->position();
+      node = node->parent();
+    }
+    if (position == node->count()) {
+      *this = save;
+    }
+  } else {
+    assert(position < node->count());
+    node = node->child(position + 1);
+    while (!node->leaf()) {
+      node = node->child(0);
+    }
+    position = 0;
+  }
+}
+
+template <typename N, typename R, typename P>
+void btree_iterator<N, R, P>::decrement_slow() {
+  if (node->leaf()) {
+    assert(position <= -1);
+    btree_iterator save(*this);
+    while (position < 0 && !node->is_root()) {
+      assert(node->parent()->child(node->position()) == node);
+      position = node->position() - 1;
+      node = node->parent();
+    }
+    if (position < 0) {
+      *this = save;
+    }
+  } else {
+    assert(position >= 0);
+    node = node->child(position);
+    while (!node->leaf()) {
+      node = node->child(node->count());
+    }
+    position = node->count() - 1;
+  }
+}
+
+////
+// btree methods
+template <typename P>
+template <typename Btree>
+void btree<P>::copy_or_move_values_in_order(Btree *x) {
+  static_assert(std::is_same_v<btree, Btree>||
+                std::is_same_v<const btree, Btree>,
+                "Btree type must be same or const.");
+  assert(empty());
+
+  // We can avoid key comparisons because we know the order of the
+  // values is the same order we'll store them in.
+  auto iter = x->begin();
+  if (iter == x->end()) return;
+  insert_multi(maybe_move_from_iterator(iter));
+  ++iter;
+  for (; iter != x->end(); ++iter) {
+    // If the btree is not empty, we can just insert the new value at the end
+    // of the tree.
+    internal_emplace(end(), maybe_move_from_iterator(iter));
+  }
+}
+
+template <typename P>
+constexpr bool btree<P>::static_assert_validation() {
+  static_assert(std::is_nothrow_copy_constructible_v<key_compare>,
+                "Key comparison must be nothrow copy constructible");
+  static_assert(std::is_nothrow_copy_constructible_v<allocator_type>,
+                "Allocator must be nothrow copy constructible");
+  static_assert(std::is_trivially_copyable_v<iterator>,
+                "iterator not trivially copyable.");
+
+  // Note: We assert that kTargetValues, which is computed from
+  // Params::kTargetNodeSize, must fit the base_fields::field_type.
+  static_assert(
+      kNodeValues < (1 << (8 * sizeof(typename node_type::field_type))),
+      "target node size too large");
+
+  // Verify that key_compare returns an absl::{weak,strong}_ordering or bool.
+  using compare_result_type =
+    std::invoke_result_t<key_compare, key_type, key_type>;
+  static_assert(
+      std::is_same_v<compare_result_type, bool> ||
+      std::is_signed_v<compare_result_type>,
+      "key comparison function must return a signed value or "
+      "bool.");
+
+  // Test the assumption made in setting kNodeValueSpace.
+  static_assert(node_type::MinimumOverhead() >= sizeof(void *) + 4,
+                "node space assumption incorrect");
+
+  return true;
+}
+
+template <typename P>
+btree<P>::btree(const key_compare &comp, const allocator_type &alloc)
+  : root_(comp, alloc, EmptyNode()), rightmost_(EmptyNode()), size_(0) {}
+
+template <typename P>
+btree<P>::btree(const btree &x) : btree(x.key_comp(), x.allocator()) {
+  copy_or_move_values_in_order(&x);
+}
+
+template <typename P>
+template <typename... Args>
+auto btree<P>::insert_unique(const key_type &key, Args &&... args)
+    -> std::pair<iterator, bool> {
+  if (empty()) {
+    mutable_root() = rightmost_ = new_leaf_root_node(1);
+  }
+
+  auto res = internal_locate(key);
+  iterator &iter = res.value;
+
+  if constexpr (res.has_match) {
+    if (res.IsEq()) {
+      // The key already exists in the tree, do nothing.
+      return {iter, false};
+    }
+  } else {
+    iterator last = internal_last(iter);
+    if (last.node && !compare_keys(key, last.key())) {
+      // The key already exists in the tree, do nothing.
+      return {last, false};
+    }
+  }
+  return {internal_emplace(iter, std::forward<Args>(args)...), true};
+}
+
+template <typename P>
+template <typename... Args>
+inline auto btree<P>::insert_hint_unique(iterator position, const key_type &key,
+                                         Args &&... args)
+    -> std::pair<iterator, bool> {
+  if (!empty()) {
+    if (position == end() || compare_keys(key, position.key())) {
+      iterator prev = position;
+      if (position == begin() || compare_keys((--prev).key(), key)) {
+        // prev.key() < key < position.key()
+        return {internal_emplace(position, std::forward<Args>(args)...), true};
+      }
+    } else if (compare_keys(position.key(), key)) {
+      ++position;
+      if (position == end() || compare_keys(key, position.key())) {
+        // {original `position`}.key() < key < {current `position`}.key()
+        return {internal_emplace(position, std::forward<Args>(args)...), true};
+      }
+    } else {
+      // position.key() == key
+      return {position, false};
+    }
+  }
+  return insert_unique(key, std::forward<Args>(args)...);
+}
+
+template <typename P>
+template <typename InputIterator>
+void btree<P>::insert_iterator_unique(InputIterator b, InputIterator e) {
+  for (; b != e; ++b) {
+    insert_hint_unique(end(), params_type::key(*b), *b);
+  }
+}
+
+template <typename P>
+template <typename ValueType>
+auto btree<P>::insert_multi(const key_type &key, ValueType&& v) -> iterator {
+  if (empty()) {
+    mutable_root() = rightmost_ = new_leaf_root_node(1);
+  }
+
+  iterator iter = internal_upper_bound(key);
+  if (iter.node == nullptr) {
+    iter = end();
+  }
+  return internal_emplace(iter, std::forward<ValueType>(v));
+}
+
+template <typename P>
+template <typename ValueType>
+auto btree<P>::insert_hint_multi(iterator position, ValueType &&v) -> iterator {
+  if (!empty()) {
+    const key_type &key = params_type::key(v);
+    if (position == end() || !compare_keys(position.key(), key)) {
+      iterator prev = position;
+      if (position == begin() || !compare_keys(key, (--prev).key())) {
+        // prev.key() <= key <= position.key()
+        return internal_emplace(position, std::forward<ValueType>(v));
+      }
+    } else {
+      iterator next = position;
+      ++next;
+      if (next == end() || !compare_keys(next.key(), key)) {
+        // position.key() < key <= next.key()
+        return internal_emplace(next, std::forward<ValueType>(v));
+      }
+    }
+  }
+  return insert_multi(std::forward<ValueType>(v));
+}
+
+template <typename P>
+template <typename InputIterator>
+void btree<P>::insert_iterator_multi(InputIterator b, InputIterator e) {
+  for (; b != e; ++b) {
+    insert_hint_multi(end(), *b);
+  }
+}
+
+template <typename P>
+auto btree<P>::operator=(const btree &x) -> btree & {
+  if (this != &x) {
+    clear();
+
+    *mutable_key_comp() = x.key_comp();
+    if constexpr (std::allocator_traits<
+                  allocator_type>::propagate_on_container_copy_assignment::value) {
+      *mutable_allocator() = x.allocator();
+    }
+
+    copy_or_move_values_in_order(&x);
+  }
+  return *this;
+}
+
+template <typename P>
+auto btree<P>::operator=(btree &&x) noexcept -> btree & {
+  if (this != &x) {
+    clear();
+
+    using std::swap;
+    if constexpr (std::allocator_traits<
+                  allocator_type>::propagate_on_container_copy_assignment::value) {
+      // Note: `root_` also contains the allocator and the key comparator.
+      swap(root_, x.root_);
+      swap(rightmost_, x.rightmost_);
+      swap(size_, x.size_);
+    } else {
+      if (allocator() == x.allocator()) {
+        swap(mutable_root(), x.mutable_root());
+        swap(*mutable_key_comp(), *x.mutable_key_comp());
+        swap(rightmost_, x.rightmost_);
+        swap(size_, x.size_);
+      } else {
+        // We aren't allowed to propagate the allocator and the allocator is
+        // different so we can't take over its memory. We must move each element
+        // individually. We need both `x` and `this` to have `x`s key comparator
+        // while moving the values so we can't swap the key comparators.
+        *mutable_key_comp() = x.key_comp();
+        copy_or_move_values_in_order(&x);
+      }
+    }
+  }
+  return *this;
+}
+
+template <typename P>
+auto btree<P>::erase(iterator iter) -> iterator {
+  bool internal_delete = false;
+  if (!iter.node->leaf()) {
+    // Deletion of a value on an internal node. First, move the largest value
+    // from our left child here, then delete that position (in remove_value()
+    // below). We can get to the largest value from our left child by
+    // decrementing iter.
+    iterator internal_iter(iter);
+    --iter;
+    assert(iter.node->leaf());
+    params_type::move(mutable_allocator(), iter.node->slot(iter.position),
+                      internal_iter.node->slot(internal_iter.position));
+    internal_delete = true;
+  }
+
+  // Delete the key from the leaf.
+  iter.node->remove_value(iter.position, mutable_allocator());
+  --size_;
+
+  // We want to return the next value after the one we just erased. If we
+  // erased from an internal node (internal_delete == true), then the next
+  // value is ++(++iter). If we erased from a leaf node (internal_delete ==
+  // false) then the next value is ++iter. Note that ++iter may point to an
+  // internal node and the value in the internal node may move to a leaf node
+  // (iter.node) when rebalancing is performed at the leaf level.
+
+  iterator res = rebalance_after_delete(iter);
+
+  // If we erased from an internal node, advance the iterator.
+  if (internal_delete) {
+    ++res;
+  }
+  return res;
+}
+
+template <typename P>
+auto btree<P>::rebalance_after_delete(iterator iter) -> iterator {
+  // Merge/rebalance as we walk back up the tree.
+  iterator res(iter);
+  bool first_iteration = true;
+  for (;;) {
+    if (iter.node == root()) {
+      try_shrink();
+      if (empty()) {
+        return end();
+      }
+      break;
+    }
+    if (iter.node->count() >= kMinNodeValues) {
+      break;
+    }
+    bool merged = try_merge_or_rebalance(&iter);
+    // On the first iteration, we should update `res` with `iter` because `res`
+    // may have been invalidated.
+    if (first_iteration) {
+      res = iter;
+      first_iteration = false;
+    }
+    if (!merged) {
+      break;
+    }
+    iter.position = iter.node->position();
+    iter.node = iter.node->parent();
+  }
+
+  // Adjust our return value. If we're pointing at the end of a node, advance
+  // the iterator.
+  if (res.position == res.node->count()) {
+    res.position = res.node->count() - 1;
+    ++res;
+  }
+
+  return res;
+}
+
+template <typename P>
+auto btree<P>::erase(iterator begin, iterator end)
+    -> std::pair<size_type, iterator> {
+  difference_type count = std::distance(begin, end);
+  assert(count >= 0);
+
+  if (count == 0) {
+    return {0, begin};
+  }
+
+  if (count == size_) {
+    clear();
+    return {count, this->end()};
+  }
+
+  if (begin.node == end.node) {
+    erase_same_node(begin, end);
+    size_ -= count;
+    return {count, rebalance_after_delete(begin)};
+  }
+
+  const size_type target_size = size_ - count;
+  while (size_ > target_size) {
+    if (begin.node->leaf()) {
+      const size_type remaining_to_erase = size_ - target_size;
+      const size_type remaining_in_node = begin.node->count() - begin.position;
+      begin = erase_from_leaf_node(
+          begin, std::min(remaining_to_erase, remaining_in_node));
+    } else {
+      begin = erase(begin);
+    }
+  }
+  return {count, begin};
+}
+
+template <typename P>
+void btree<P>::erase_same_node(iterator begin, iterator end) {
+  assert(begin.node == end.node);
+  assert(end.position > begin.position);
+
+  node_type *node = begin.node;
+  size_type to_erase = end.position - begin.position;
+  if (!node->leaf()) {
+    // Delete all children between begin and end.
+    for (size_type i = 0; i < to_erase; ++i) {
+      internal_clear(node->child(begin.position + i + 1));
+    }
+    // Rotate children after end into new positions.
+    for (size_type i = begin.position + to_erase + 1; i <= node->count(); ++i) {
+      node->set_child(i - to_erase, node->child(i));
+      node->clear_child(i);
+    }
+  }
+  node->remove_values_ignore_children(begin.position, to_erase,
+                                      mutable_allocator());
+
+  // Do not need to update rightmost_, because
+  // * either end == this->end(), and therefore node == rightmost_, and still
+  //   exists
+  // * or end != this->end(), and therefore rightmost_ hasn't been erased, since
+  //   it wasn't covered in [begin, end)
+}
+
+template <typename P>
+auto btree<P>::erase_from_leaf_node(iterator begin, size_type to_erase)
+    -> iterator {
+  node_type *node = begin.node;
+  assert(node->leaf());
+  assert(node->count() > begin.position);
+  assert(begin.position + to_erase <= node->count());
+
+  node->remove_values_ignore_children(begin.position, to_erase,
+                                      mutable_allocator());
+
+  size_ -= to_erase;
+
+  return rebalance_after_delete(begin);
+}
+
+template <typename P>
+template <typename K>
+auto btree<P>::erase_unique(const K &key) -> size_type {
+  const iterator iter = internal_find(key);
+  if (iter.node == nullptr) {
+    // The key doesn't exist in the tree, return nothing done.
+    return 0;
+  }
+  erase(iter);
+  return 1;
+}
+
+template <typename P>
+template <typename K>
+auto btree<P>::erase_multi(const K &key) -> size_type {
+  const iterator begin = internal_lower_bound(key);
+  if (begin.node == nullptr) {
+    // The key doesn't exist in the tree, return nothing done.
+    return 0;
+  }
+  // Delete all of the keys between begin and upper_bound(key).
+  const iterator end = internal_end(internal_upper_bound(key));
+  return erase(begin, end).first;
+}
+
+template <typename P>
+void btree<P>::clear() {
+  if (!empty()) {
+    internal_clear(root());
+  }
+  mutable_root() = EmptyNode();
+  rightmost_ = EmptyNode();
+  size_ = 0;
+}
+
+template <typename P>
+void btree<P>::swap(btree &x) {
+  using std::swap;
+  if (std::allocator_traits<
+          allocator_type>::propagate_on_container_swap::value) {
+    // Note: `root_` also contains the allocator and the key comparator.
+    swap(root_, x.root_);
+  } else {
+    // It's undefined behavior if the allocators are unequal here.
+    assert(allocator() == x.allocator());
+    swap(mutable_root(), x.mutable_root());
+    swap(*mutable_key_comp(), *x.mutable_key_comp());
+  }
+  swap(rightmost_, x.rightmost_);
+  swap(size_, x.size_);
+}
+
+template <typename P>
+void btree<P>::verify() const {
+  assert(root() != nullptr);
+  assert(leftmost() != nullptr);
+  assert(rightmost_ != nullptr);
+  assert(empty() || size() == internal_verify(root(), nullptr, nullptr));
+  assert(leftmost() == (++const_iterator(root(), -1)).node);
+  assert(rightmost_ == (--const_iterator(root(), root()->count())).node);
+  assert(leftmost()->leaf());
+  assert(rightmost_->leaf());
+}
+
+template <typename P>
+void btree<P>::rebalance_or_split(iterator *iter) {
+  node_type *&node = iter->node;
+  int &insert_position = iter->position;
+  assert(node->count() == node->max_count());
+  assert(kNodeValues == node->max_count());
+
+  // First try to make room on the node by rebalancing.
+  node_type *parent = node->parent();
+  if (node != root()) {
+    if (node->position() > 0) {
+      // Try rebalancing with our left sibling.
+      node_type *left = parent->child(node->position() - 1);
+      assert(left->max_count() == kNodeValues);
+      if (left->count() < kNodeValues) {
+        // We bias rebalancing based on the position being inserted. If we're
+        // inserting at the end of the right node then we bias rebalancing to
+        // fill up the left node.
+        int to_move = (kNodeValues - left->count()) /
+                      (1 + (insert_position < kNodeValues));
+        to_move = std::max(1, to_move);
+
+        if (((insert_position - to_move) >= 0) ||
+            ((left->count() + to_move) < kNodeValues)) {
+          left->rebalance_right_to_left(to_move, node, mutable_allocator());
+
+          assert(node->max_count() - node->count() == to_move);
+          insert_position = insert_position - to_move;
+          if (insert_position < 0) {
+            insert_position = insert_position + left->count() + 1;
+            node = left;
+          }
+
+          assert(node->count() < node->max_count());
+          return;
+        }
+      }
+    }
+
+    if (node->position() < parent->count()) {
+      // Try rebalancing with our right sibling.
+      node_type *right = parent->child(node->position() + 1);
+      assert(right->max_count() == kNodeValues);
+      if (right->count() < kNodeValues) {
+        // We bias rebalancing based on the position being inserted. If we're
+        // inserting at the beginning of the left node then we bias rebalancing
+        // to fill up the right node.
+        int to_move =
+            (kNodeValues - right->count()) / (1 + (insert_position > 0));
+        to_move = (std::max)(1, to_move);
+
+        if ((insert_position <= (node->count() - to_move)) ||
+            ((right->count() + to_move) < kNodeValues)) {
+          node->rebalance_left_to_right(to_move, right, mutable_allocator());
+
+          if (insert_position > node->count()) {
+            insert_position = insert_position - node->count() - 1;
+            node = right;
+          }
+
+          assert(node->count() < node->max_count());
+          return;
+        }
+      }
+    }
+
+    // Rebalancing failed, make sure there is room on the parent node for a new
+    // value.
+    assert(parent->max_count() == kNodeValues);
+    if (parent->count() == kNodeValues) {
+      iterator parent_iter(node->parent(), node->position());
+      rebalance_or_split(&parent_iter);
+    }
+  } else {
+    // Rebalancing not possible because this is the root node.
+    // Create a new root node and set the current root node as the child of the
+    // new root.
+    parent = new_internal_node(parent);
+    parent->init_child(0, root());
+    mutable_root() = parent;
+    // If the former root was a leaf node, then it's now the rightmost node.
+    assert(!parent->child(0)->leaf() || parent->child(0) == rightmost_);
+  }
+
+  // Split the node.
+  node_type *split_node;
+  if (node->leaf()) {
+    split_node = new_leaf_node(parent);
+    node->split(insert_position, split_node, mutable_allocator());
+    if (rightmost_ == node) rightmost_ = split_node;
+  } else {
+    split_node = new_internal_node(parent);
+    node->split(insert_position, split_node, mutable_allocator());
+  }
+
+  if (insert_position > node->count()) {
+    insert_position = insert_position - node->count() - 1;
+    node = split_node;
+  }
+}
+
+template <typename P>
+void btree<P>::merge_nodes(node_type *left, node_type *right) {
+  left->merge(right, mutable_allocator());
+  if (right->leaf()) {
+    if (rightmost_ == right) rightmost_ = left;
+    delete_leaf_node(right);
+  } else {
+    delete_internal_node(right);
+  }
+}
+
+template <typename P>
+bool btree<P>::try_merge_or_rebalance(iterator *iter) {
+  node_type *parent = iter->node->parent();
+  if (iter->node->position() > 0) {
+    // Try merging with our left sibling.
+    node_type *left = parent->child(iter->node->position() - 1);
+    assert(left->max_count() == kNodeValues);
+    if ((1 + left->count() + iter->node->count()) <= kNodeValues) {
+      iter->position += 1 + left->count();
+      merge_nodes(left, iter->node);
+      iter->node = left;
+      return true;
+    }
+  }
+  if (iter->node->position() < parent->count()) {
+    // Try merging with our right sibling.
+    node_type *right = parent->child(iter->node->position() + 1);
+    assert(right->max_count() == kNodeValues);
+    if ((1 + iter->node->count() + right->count()) <= kNodeValues) {
+      merge_nodes(iter->node, right);
+      return true;
+    }
+    // Try rebalancing with our right sibling. We don't perform rebalancing if
+    // we deleted the first element from iter->node and the node is not
+    // empty. This is a small optimization for the common pattern of deleting
+    // from the front of the tree.
+    if ((right->count() > kMinNodeValues) &&
+        ((iter->node->count() == 0) ||
+         (iter->position > 0))) {
+      int to_move = (right->count() - iter->node->count()) / 2;
+      to_move = std::min(to_move, right->count() - 1);
+      iter->node->rebalance_right_to_left(to_move, right, mutable_allocator());
+      return false;
+    }
+  }
+  if (iter->node->position() > 0) {
+    // Try rebalancing with our left sibling. We don't perform rebalancing if
+    // we deleted the last element from iter->node and the node is not
+    // empty. This is a small optimization for the common pattern of deleting
+    // from the back of the tree.
+    node_type *left = parent->child(iter->node->position() - 1);
+    if ((left->count() > kMinNodeValues) &&
+        ((iter->node->count() == 0) ||
+         (iter->position < iter->node->count()))) {
+      int to_move = (left->count() - iter->node->count()) / 2;
+      to_move = std::min(to_move, left->count() - 1);
+      left->rebalance_left_to_right(to_move, iter->node, mutable_allocator());
+      iter->position += to_move;
+      return false;
+    }
+  }
+  return false;
+}
+
+template <typename P>
+void btree<P>::try_shrink() {
+  if (root()->count() > 0) {
+    return;
+  }
+  // Deleted the last item on the root node, shrink the height of the tree.
+  if (root()->leaf()) {
+    assert(size() == 0);
+    delete_leaf_node(root());
+    mutable_root() = EmptyNode();
+    rightmost_ = EmptyNode();
+  } else {
+    node_type *child = root()->child(0);
+    child->make_root();
+    delete_internal_node(root());
+    mutable_root() = child;
+  }
+}
+
+template <typename P>
+template <typename IterType>
+inline IterType btree<P>::internal_last(IterType iter) {
+  assert(iter.node != nullptr);
+  while (iter.position == iter.node->count()) {
+    iter.position = iter.node->position();
+    iter.node = iter.node->parent();
+    if (iter.node->leaf()) {
+      iter.node = nullptr;
+      break;
+    }
+  }
+  return iter;
+}
+
+template <typename P>
+template <typename... Args>
+inline auto btree<P>::internal_emplace(iterator iter, Args &&... args)
+    -> iterator {
+  if (!iter.node->leaf()) {
+    // We can't insert on an internal node. Instead, we'll insert after the
+    // previous value which is guaranteed to be on a leaf node.
+    --iter;
+    ++iter.position;
+  }
+  const int max_count = iter.node->max_count();
+  if (iter.node->count() == max_count) {
+    // Make room in the leaf for the new item.
+    if (max_count < kNodeValues) {
+      // Insertion into the root where the root is smaller than the full node
+      // size. Simply grow the size of the root node.
+      assert(iter.node == root());
+      iter.node =
+          new_leaf_root_node(std::min(kNodeValues, 2 * max_count));
+      iter.node->swap(root(), mutable_allocator());
+      delete_leaf_node(root());
+      mutable_root() = iter.node;
+      rightmost_ = iter.node;
+    } else {
+      rebalance_or_split(&iter);
+    }
+  }
+  iter.node->emplace_value(iter.position, mutable_allocator(),
+                           std::forward<Args>(args)...);
+  ++size_;
+  return iter;
+}
+
+template <typename P>
+template <typename K>
+inline auto btree<P>::internal_locate(const K &key) const
+    -> SearchResult<iterator, is_key_compare_to::value> {
+  return internal_locate_impl(key, is_key_compare_to());
+}
+
+template <typename P>
+template <typename K>
+inline auto btree<P>::internal_locate_impl(
+    const K &key, std::false_type /* IsCompareTo */) const
+    -> SearchResult<iterator, false> {
+  iterator iter(const_cast<node_type *>(root()), 0);
+  for (;;) {
+    iter.position = iter.node->lower_bound(key, key_comp()).value;
+    // NOTE: we don't need to walk all the way down the tree if the keys are
+    // equal, but determining equality would require doing an extra comparison
+    // on each node on the way down, and we will need to go all the way to the
+    // leaf node in the expected case.
+    if (iter.node->leaf()) {
+      break;
+    }
+    iter.node = iter.node->child(iter.position);
+  }
+  return {iter};
+}
+
+template <typename P>
+template <typename K>
+inline auto btree<P>::internal_locate_impl(
+    const K &key, std::true_type /* IsCompareTo */) const
+    -> SearchResult<iterator, true> {
+  iterator iter(const_cast<node_type *>(root()), 0);
+  for (;;) {
+    SearchResult<int, true> res = iter.node->lower_bound(key, key_comp());
+    iter.position = res.value;
+    if (res.match == MatchKind::kEq) {
+      return {iter, MatchKind::kEq};
+    }
+    if (iter.node->leaf()) {
+      break;
+    }
+    iter.node = iter.node->child(iter.position);
+  }
+  return {iter, MatchKind::kNe};
+}
+
+template <typename P>
+template <typename K>
+auto btree<P>::internal_lower_bound(const K &key) const -> iterator {
+  iterator iter(const_cast<node_type *>(root()), 0);
+  for (;;) {
+    iter.position = iter.node->lower_bound(key, key_comp()).value;
+    if (iter.node->leaf()) {
+      break;
+    }
+    iter.node = iter.node->child(iter.position);
+  }
+  return internal_last(iter);
+}
+
+template <typename P>
+template <typename K>
+auto btree<P>::internal_upper_bound(const K &key) const -> iterator {
+  iterator iter(const_cast<node_type *>(root()), 0);
+  for (;;) {
+    iter.position = iter.node->upper_bound(key, key_comp());
+    if (iter.node->leaf()) {
+      break;
+    }
+    iter.node = iter.node->child(iter.position);
+  }
+  return internal_last(iter);
+}
+
+template <typename P>
+template <typename K>
+auto btree<P>::internal_find(const K &key) const -> iterator {
+  auto res = internal_locate(key);
+  if constexpr (res.has_match) {
+    if (res.IsEq()) {
+      return res.value;
+    }
+  } else {
+    const iterator iter = internal_last(res.value);
+    if (iter.node != nullptr && !compare_keys(key, iter.key())) {
+      return iter;
+    }
+  }
+  return {nullptr, 0};
+}
+
+template <typename P>
+void btree<P>::internal_clear(node_type *node) {
+  if (!node->leaf()) {
+    for (int i = 0; i <= node->count(); ++i) {
+      internal_clear(node->child(i));
+    }
+    delete_internal_node(node);
+  } else {
+    delete_leaf_node(node);
+  }
+}
+
+template <typename P>
+int btree<P>::internal_verify(
+    const node_type *node, const key_type *lo, const key_type *hi) const {
+  assert(node->count() > 0);
+  assert(node->count() <= node->max_count());
+  if (lo) {
+    assert(!compare_keys(node->key(0), *lo));
+  }
+  if (hi) {
+    assert(!compare_keys(*hi, node->key(node->count() - 1)));
+  }
+  for (int i = 1; i < node->count(); ++i) {
+    assert(!compare_keys(node->key(i), node->key(i - 1)));
+  }
+  int count = node->count();
+  if (!node->leaf()) {
+    for (int i = 0; i <= node->count(); ++i) {
+      assert(node->child(i) != nullptr);
+      assert(node->child(i)->parent() == node);
+      assert(node->child(i)->position() == i);
+      count += internal_verify(
+          node->child(i),
+          (i == 0) ? lo : &node->key(i - 1),
+          (i == node->count()) ? hi : &node->key(i));
+    }
+  }
+  return count;
+}
+
+} // namespace btree::internal