// Copyright 2016 The Fuchsia Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.

#ifndef FBL_INTRUSIVE_WAVL_TREE_H_
#define FBL_INTRUSIVE_WAVL_TREE_H_

#include <zircon/assert.h>

#include <cstddef>
#include <iterator>
#include <utility>

#include <fbl/algorithm.h>
#include <fbl/intrusive_container_node_utils.h>
#include <fbl/intrusive_container_utils.h>
#include <fbl/intrusive_pointer_traits.h>
#include <fbl/intrusive_wavl_tree_internal.h>

// Implementation Notes:
//
// WAVLTree<> is an implementation of a "Weak AVL" tree; a self
// balancing binary search tree whose rebalancing algorithm was
// originally described in
//
// Bernhard Haeupler, Siddhartha Sen, and Robert E. Tarjan. 2015.
// Rank-Balanced Trees. ACM Trans. Algorithms 11, 4, Article 30 (June 2015), 26 pages.
// DOI=http://dx.doi.org/10.1145/2689412
//
// See also
// https://en.wikipedia.org/wiki/WAVL_tree
// http://sidsen.azurewebsites.net/papers/rb-trees-talg.pdf
//
// WAVLTree<>s, like HashTables, are associative containers and support all of
// the same key-centric operations (such as find() and insert_or_find()) that
// HashTables support.
//
// Additionally, WAVLTree's are internally ordered by key (unlike HashTables
// which are un-ordered).  Iteration forwards or backwards runs in amortized
// constant time, but in O(log) time in an individual worst case.  Forward
// iteration will enumerate the elements in monotonically increasing order (as
// defined by the KeyTraits::LessThan operation).
//
// Two additional operations are supported because of the ordered nature of a
// WAVLTree:
// upper_bound(key) : Finds the element (E) in the tree such that E.key > key
// lower_bound(key) : Finds the element (E) in the tree such that E.key >= key
//
// The worst depth of a WAVL tree depends on whether or not the tree has ever
// been subject to erase operations.
// ++ If the tree has seen only insert operations, the worst case depth of the
//    tree is log_phi(N), where phi is the golden ratio.  This is the same bound
//    as that of an AVL tree.
// ++ If the tree has seen erase operations in addition to insert operations,
//    the worst case depth of the tree is 2*log_2(N).  This is the same bound as
//    a Red-Black tree.
//
// Insertion runs in O(log) time; finding the location takes O(log) time while
// post-insert rebalancing runs in amortized constant time.
//
// Erase-by-key runs in O(log) time; finding the node to erase takes O(log) time
// while post-erase rebalancing runs in amortized constant time.
//
// Because of the intrusive nature of the container, direct-erase operations
// (AKA, erase operations where the reference to the element to be erased is
// already known) run in amortized constant time.
//
namespace fbl {

namespace tests {
namespace intrusive_containers {
class WAVLTreeChecker;
class WAVLBalanceTestObserver;
}  // namespace intrusive_containers
}  // namespace tests

template <typename PtrType_, NodeOptions Options, typename RankType>
struct WAVLTreeNodeStateBase
    : public internal::CommonNodeStateBase<WAVLTreeNodeStateBase<PtrType_, Options, RankType>> {
 private:
  using Base = internal::CommonNodeStateBase<WAVLTreeNodeStateBase<PtrType_, Options, RankType>>;

 public:
  using PtrType = PtrType_;
  using PtrTraits = internal::ContainerPtrTraits<PtrType_>;
  static constexpr NodeOptions kNodeOptions = Options;

  WAVLTreeNodeStateBase() = default;

  ~WAVLTreeNodeStateBase() {
    ZX_DEBUG_ASSERT(IsValid());
    if constexpr (!(kNodeOptions & fbl::NodeOptions::AllowClearUnsafe)) {
      ZX_DEBUG_ASSERT(!InContainer());
    }
  }

  bool IsValid() const { return (parent_ || (!parent_ && !left_ && !right_)); }
  bool InContainer() const { return (parent_ != nullptr); }

  // Defer to CommonNodeStateBase for enforcement of the various copy/move
  // rules.  Make sure, however, that we explicitly do not allow our own default
  // construction/assignment operators change anything about our state.
  WAVLTreeNodeStateBase(const WAVLTreeNodeStateBase& other) : Base(other) {}
  WAVLTreeNodeStateBase& operator=(const WAVLTreeNodeStateBase& other) {
    this->Base::operator=(other);
    return *this;
  }
  WAVLTreeNodeStateBase(WAVLTreeNodeStateBase&& other) : Base(std::move(other)) {}
  WAVLTreeNodeStateBase& operator=(WAVLTreeNodeStateBase&& other) {
    this->Base::operator=(std::move(other));
    return *this;
  }

 protected:
  template <typename, typename, typename, typename, typename, typename>
  friend class WAVLTree;
  friend class tests::intrusive_containers::WAVLTreeChecker;
  friend class tests::intrusive_containers::WAVLBalanceTestObserver;

  typename PtrTraits::RawPtrType parent_ = nullptr;
  typename PtrTraits::RawPtrType left_ = nullptr;
  typename PtrTraits::RawPtrType right_ = nullptr;
  RankType rank_{};
};

template <typename PtrType_, NodeOptions Options>
struct WAVLTreeNodeState<PtrType_, Options, DefaultWAVLTreeRankType>
    : public WAVLTreeNodeStateBase<PtrType_, Options, DefaultWAVLTreeRankType> {
  WAVLTreeNodeState() = default;

  // Defer to WAVLTreeNodeStateBase for enforcement of the various copy/move rules.
  WAVLTreeNodeState(const WAVLTreeNodeState&) = default;
  WAVLTreeNodeState& operator=(const WAVLTreeNodeState&) = default;
  WAVLTreeNodeState(WAVLTreeNodeState&&) = default;
  WAVLTreeNodeState& operator=(WAVLTreeNodeState&&) = default;

  bool rank_parity() const { return this->rank_; }
  void promote_rank() { this->rank_ = !this->rank_; }
  void double_promote_rank() {}
  void demote_rank() { this->rank_ = !this->rank_; }
  void double_demote_rank() {}
};

template <typename PtrType, NodeOptions Options, typename TagType>
struct WAVLTreeContainable;

template <typename PtrType_, typename TagType_>
struct DefaultWAVLTreeTraits {
 private:
  using ValueType = typename internal::ContainerPtrTraits<PtrType_>::ValueType;

 public:
  using PtrType = PtrType_;
  using TagType = TagType_;
  using PtrTraits = internal::ContainerPtrTraits<PtrType_>;

  static auto& node_state(typename PtrTraits::RefType obj) {
    if constexpr (std::is_same_v<TagType, DefaultObjectTag>) {
      return obj.ValueType::wavl_node_state_;
    } else {
      return obj.template GetContainableByTag<TagType>().wavl_node_state_;
    }
  }

  using NodeState =
      std::decay_t<std::invoke_result_t<decltype(node_state), typename PtrTraits::RefType>>;
};

template <typename PtrType_, NodeOptions Options = NodeOptions::None,
          typename TagType_ = DefaultObjectTag>
struct WAVLTreeContainable {
 public:
  using PtrType = PtrType_;
  using TagType = TagType_;

  bool InContainer() const {
    using Node = WAVLTreeContainable<PtrType, Options, TagType>;
    return Node::wavl_node_state_.InContainer();
  }

 private:
  friend DefaultWAVLTreeTraits<PtrType, TagType>;
  WAVLTreeNodeState<PtrType, Options, DefaultWAVLTreeRankType> wavl_node_state_;
};

template <typename KeyType_, typename PtrType_,
          typename KeyTraits_ = DefaultKeyedObjectTraits<
              KeyType_, typename internal::ContainerPtrTraits<PtrType_>::ValueType>,
          typename TagType_ = DefaultObjectTag,
          typename NodeTraits_ = DefaultWAVLTreeTraits<PtrType_, TagType_>,
          typename Observer_ = tests::intrusive_containers::DefaultWAVLTreeObserver>
class __POINTER(KeyType_) WAVLTree {
 private:
  // Private fwd decls of the iterator implementation.
  template <typename IterTraits>
  class iterator_impl;
  struct iterator_traits;
  struct const_iterator_traits;

 public:
  // Aliases used to reduce verbosity and expose types/traits to tests
  using KeyType = KeyType_;
  using PtrType = PtrType_;
  using KeyTraits = KeyTraits_;
  using TagType = TagType_;
  using NodeTraits = NodeTraits_;
  using Observer = Observer_;

  using PtrTraits = internal::ContainerPtrTraits<PtrType>;
  using RawPtrType = typename PtrTraits::RawPtrType;
  using ValueType = typename PtrTraits::ValueType;
  using RefType = typename PtrTraits::RefType;
  using ContainerType = WAVLTree<KeyType_, PtrType_, KeyTraits_, TagType_, NodeTraits_, Observer_>;
  using CheckerType = ::fbl::tests::intrusive_containers::WAVLTreeChecker;

  // Declarations of the standard iterator types.
  using iterator = iterator_impl<iterator_traits>;
  using const_iterator = iterator_impl<const_iterator_traits>;

  // WAVL Trees support amortized constant erase.  Technically, the worst case
  // for any individual erase operation involves O(log) demotions, followed by
  // a double rotation operation.  Given D total erase operations, however,
  // the maximum number of operations (demotions + rotations) is 2*D, given
  // the amortized constant erase time.
  //
  static constexpr bool SupportsConstantOrderErase = true;
  static constexpr bool SupportsConstantOrderSize = true;
  static constexpr bool IsAssociative = true;
  static constexpr bool IsSequenced = false;

  // Default construction gives an empty tree.
  constexpr WAVLTree() noexcept {
    using NodeState = internal::node_state_t<NodeTraits, RefType>;

    // Make certain that the type of pointer we are expected to manage matches
    // the type of pointer that our Node type expects to manage.
    static_assert(std::is_same_v<PtrType, typename NodeState::PtrType>,
                  "WAVLTree's pointer type must match its Node's pointerType");

    // WAVLTree does not currently support direct remove-from-container.
    static_assert(!(NodeState::kNodeOptions & NodeOptions::AllowRemoveFromContainer),
                  "WAVLTree does not support nodes which allow RemoveFromContainer.");
  }

  // Rvalue construction is permitted, but will result in the move of the tree
  // contents from one instance of the list to the other (even for unmanaged
  // pointers)
  WAVLTree(WAVLTree&& other_tree) noexcept : WAVLTree() { swap(other_tree); }

  // Rvalue assignment is permitted for managed trees, and when the target is
  // an empty tree of unmanaged pointers.  Like Rvalue construction, it will
  // result in the move of the source contents to the destination.
  WAVLTree& operator=(WAVLTree&& other_tree) {
    ZX_DEBUG_ASSERT(PtrTraits::IsManaged || is_empty());

    clear();
    swap(other_tree);

    return *this;
  }

  ~WAVLTree() {
    // It is considered an error to allow a tree of unmanaged pointers to
    // destruct of there are still elements in it.  Managed pointer trees
    // will automatically release their references to their elements.
    ZX_DEBUG_ASSERT(PtrTraits::IsManaged || is_empty());
    clear();
  }

  // Standard begin/end, cbegin/cend iterator accessors.
  iterator begin() { return iterator(left_most_); }
  const_iterator begin() const { return const_iterator(left_most_); }
  const_iterator cbegin() const { return const_iterator(left_most_); }

  iterator end() { return iterator(sentinel()); }
  const_iterator end() const { return const_iterator(sentinel()); }
  const_iterator cend() const { return const_iterator(sentinel()); }

  // Iterator accessors to the root node.
  iterator root() { return is_empty() ? end() : iterator(root_); }
  const_iterator root() const { return is_empty() ? cend() : const_iterator(root_); }
  const_iterator croot() const { return is_empty() ? cend() : const_iterator(root_); }

  // make_iterator : construct an iterator out of a pointer to an object
  iterator make_iterator(ValueType& obj) { return iterator(&obj); }
  const_iterator make_iterator(const ValueType& obj) const {
    return const_iterator(&const_cast<ValueType&>(obj));
  }

  // is_empty : True if the tree has at least one element in it, false otherwise.
  bool is_empty() const { return root_ == nullptr; }

  // front
  //
  // Return a reference to the element at the front of the list without
  // removing it.  It is an error to call front on an empty list.
  typename PtrTraits::RefType front() {
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(left_most_));
    return *left_most_;
  }

  typename PtrTraits::ConstRefType front() const {
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(left_most_));
    return *left_most_;
  }

  // back
  //
  // Return a reference to the element at the back of the list without
  // removing it.  It is an error to call back on an empty list.
  typename PtrTraits::RefType back() {
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(right_most_));
    return *right_most_;
  }

  typename PtrTraits::ConstRefType back() const {
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(right_most_));
    return *right_most_;
  }

  void insert(const PtrType& ptr) { insert(PtrType(ptr)); }
  void insert(PtrType&& ptr) { internal_insert(ptr); }

  // insert or find
  //
  // Insert the object pointed to by ptr if it is not already in the
  // tree, or find the object that the ptr collided with instead.
  //
  // @param ptr A pointer to the new object to insert.
  // @param iter An optional out parameter pointer to an iterator which
  // will reference either the newly inserted item, or the item whose key
  // collided with ptr.
  //
  // @return true if there was no collision and the item was successfully
  // inserted.  false otherwise.
  //
  bool insert_or_find(const PtrType& ptr, iterator* iter = nullptr) {
    return insert_or_find(PtrType(ptr), iter);
  }

  bool insert_or_find(PtrType&& ptr, iterator* iter = nullptr) {
    RawPtrType obj = PtrTraits::GetRaw(ptr);
    RawPtrType collision = nullptr;

    internal_insert(ptr, &collision);

    if (collision) {
      Observer::RecordInsertCollision(obj, iterator(collision));
    }

    if (iter) {
      *iter = collision ? iterator(collision) : iterator(obj);
    }

    return !collision;
  }

  // insert_or_replace
  //
  // Find the element in the tree with the same key as *ptr and replace
  // it with ptr, then return the pointer to the element which was replaced.
  // If no element in the tree shares a key with *ptr, simply add ptr to
  // the tree and return nullptr.
  //
  PtrType insert_or_replace(const PtrType& ptr) { return insert_or_replace(PtrType(ptr)); }

  PtrType insert_or_replace(PtrType&& ptr) {
    ZX_DEBUG_ASSERT(ptr != nullptr);
    ZX_DEBUG_ASSERT(!NodeTraits::node_state(*ptr).InContainer());

    RawPtrType collision = nullptr;
    internal_insert(ptr, &collision);

    // If there was a collision, swap our node with the node we collided
    // with.
    if (collision) {
      ZX_DEBUG_ASSERT(ptr != nullptr);
      Observer::RecordInsertReplace(iterator(collision), PtrTraits::GetRaw(ptr));
      return internal_swap(collision, std::move(ptr));
    }

    return nullptr;
  }

  // pop_front and pop_back
  //
  // Removes either the left-most or right-most member of tree and transfers
  // the pointer to the caller.  If the list is empty, return a nullptr
  // instance of PtrType.
  PtrType pop_front() { return internal_erase(left_most_); }
  PtrType pop_back() { return internal_erase(right_most_); }

  // find
  //
  // Find the first node in the tree whose key matches "key" and return an
  // iterator to it.  Return end() if no node in the tree has a key which
  // matches "key".
  const_iterator find(const KeyType& key) const {
    RawPtrType node = root_;

    while (internal::valid_sentinel_ptr(node)) {
      auto node_key = KeyTraits::GetKey(*node);

      if (KeyTraits::EqualTo(key, node_key))
        return const_iterator(node);

      auto& ns = NodeTraits::node_state(*node);
      node = KeyTraits::LessThan(key, node_key) ? ns.left_ : ns.right_;
    }

    return end();
  }

  iterator find(const KeyType& key) {
    const_iterator citer = const_cast<const ContainerType*>(this)->find(key);
    return iterator(citer.node_);
  }

  // upper_bound
  //
  // Find the first node in the tree whose key is strictly greater than the
  // caller provided key.  Returns end() if no such node exists.
  const_iterator upper_bound(const KeyType& key) const {
    return internal_upper_lower_bound<UpperBoundTraits>(key);
  }

  iterator upper_bound(const KeyType& key) {
    const_iterator citer = const_cast<const ContainerType*>(this)->upper_bound(key);
    return iterator(citer.node_);
  }

  // lower_bound
  //
  // Find the first node in the tree whose key is greater than or equal to the
  // caller provided key.  Returns end() if no such node exists.
  const_iterator lower_bound(const KeyType& key) const {
    return internal_upper_lower_bound<LowerBoundTraits>(key);
  }

  iterator lower_bound(const KeyType& key) {
    const_iterator citer = const_cast<const ContainerType*>(this)->lower_bound(key);
    return iterator(citer.node_);
  }

  // erase
  //
  // Remove the first element in the tree whose key matches "key" and return a
  // pointer the removed object.  Return a nullptr instance of PtrType if no
  // such element exists in the tree.
  PtrType erase(const KeyType& key) { return erase(find(key)); }

  // erase (direct)
  //
  // Remove the object directly referenced either by "iter" or "obj" from the
  // tree and return a pointer to it.  In the case of an iterator based erase,
  // return a nullptr instance of PtrType if the iterator is invalid.  It is
  // an error to either use a valid iterator from a different tree instance,
  // or to attempt to remove an element which is not currently a member of
  // this tree instance.
  PtrType erase(const iterator& iter) {
    if (!iter.IsValid())
      return PtrType(nullptr);

    return internal_erase(&(*iter));
  }

  PtrType erase(ValueType& obj) { return internal_erase(&obj); }

  // clear
  //
  // Clear out the tree, unlinking all of the elements in the process.  For
  // managed pointer types, this will release all references held by the tree
  // to the objects which were in it.
  void clear() {
    if (is_empty())
      return;

    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(root_));
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(left_most_));
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(right_most_));

    // Clear the left and right sentinels right now so that we don't have
    // to worry about special casing them while cleaning up the tree.
    NodeTraits::node_state(*left_most_).left_ = nullptr;
    NodeTraits::node_state(*right_most_).right_ = nullptr;

    RawPtrType owner = sentinel();
    RawPtrType* link_ptr = &root_;

    while (true) {
      auto& ns = NodeTraits::node_state(**link_ptr);

      if ((ns.left_ == nullptr) && (ns.right_ == nullptr)) {
        // Leaf node.  Trim it.  Start by reclaiming the pointer
        // reference (if this is a managed pointer type) and holding
        // onto it until the end of this scope.  Note, we need to flag
        // this temp copy of the reference as UNUSED in case the pointer
        // type is unmanaged.
        ZX_DEBUG_ASSERT(ns.parent_ == owner);
        PtrType leaf = PtrTraits::Reclaim(*link_ptr);

        // This leaf node node now has no parent, and the pointer which
        // pointed to us is now null.
        ns.parent_ = nullptr;
        *link_ptr = nullptr;

        // Now that the node has been removed from the tree, inform any observer
        // that the node has been cleared by calling RecordErase, but do not
        // provide any "invalidated" node iterator which exists up-tree of the
        // node which was just removed.  The entire tree is being cleared, so
        // any augmented invariant the tree is maintaining is about to become
        // undefined anyway.
        Observer::RecordErase(PtrTraits::GetRaw(leaf), iterator());

        // If we are removing the root, and it is a leaf node, then we
        // are done.
        if (link_ptr == &root_)
          break;

        // Now climb back up the tree.
        link_ptr = &GetLinkPtrToNode(owner);
        owner = NodeTraits::node_state(*owner).parent_;
      } else {
        // Non-leaf node, descend.  We have already detached the left
        // and right sentinels, so we shouldn't be seeing any here.
        ZX_DEBUG_ASSERT(!internal::is_sentinel_ptr(ns.left_));
        ZX_DEBUG_ASSERT(!internal::is_sentinel_ptr(ns.right_));

        owner = *link_ptr;
        link_ptr = (ns.left_ != nullptr) ? &ns.left_ : &ns.right_;
      }
    }

    ZX_DEBUG_ASSERT(root_ == nullptr);
    left_most_ = sentinel();
    right_most_ = sentinel();
    count_ = 0;
  }

  // clear_unsafe
  //
  // See comments in fbl/intrusive_single_list.h
  // Think carefully before calling this!
  void clear_unsafe() {
    static_assert(PtrTraits::IsManaged == false,
                  "clear_unsafe is not allowed for containers of managed pointers");
    static_assert(NodeTraits::NodeState::kNodeOptions & NodeOptions::AllowClearUnsafe,
                  "Container does not support clear_unsafe.  Consider adding "
                  "NodeOptions::AllowClearUnsafe to your node storage.");

    root_ = nullptr;
    left_most_ = sentinel();
    right_most_ = sentinel();
    count_ = 0;
  }

  // swap : swaps the contents of two trees.
  void swap(WAVLTree& other) {
    internal::Swap(root_, other.root_);
    internal::Swap(left_most_, other.left_most_);
    internal::Swap(right_most_, other.right_most_);
    internal::Swap(count_, other.count_);

    // Fix up the sentinel values.
    FixSentinelsAfterSwap();
    other.FixSentinelsAfterSwap();
  }

  // size : return the current number of elements in the tree.
  size_t size() const { return count_; }

  // erase_if
  //
  // Find the first member of the list which satisfies the predicate given by
  // 'fn' and erase it from the list, returning a referenced pointer to the
  // removed element.  Return nullptr if no element satisfies the predicate.
  template <typename UnaryFn>
  PtrType erase_if(UnaryFn fn) {
    for (auto iter = begin(); iter != end(); ++iter) {
      if (fn(*iter))
        return erase(iter);
    }
    return PtrType(nullptr);
  }

  // find_if
  //
  // Find the first member of the list which satisfies the predicate given by
  // 'fn' and return a const& to the PtrType in the list which refers to it.
  // Return nullptr if no member satisfies the predicate.
  template <typename UnaryFn>
  const_iterator find_if(UnaryFn fn) const {
    for (auto iter = begin(); iter != end(); ++iter)
      if (fn(*iter))
        return const_iterator(iter.node_);

    return const_iterator(sentinel());
  }

  template <typename UnaryFn>
  iterator find_if(UnaryFn fn) {
    const_iterator citer = const_cast<const ContainerType*>(this)->find_if(fn);
    return iterator(citer.node_);
  }

 private:
  // The traits of a non-const iterator
  struct iterator_traits {
    using RefType = typename PtrTraits::RefType;
    using RawPtrType = typename PtrTraits::RawPtrType;
  };

  // The traits of a const iterator
  struct const_iterator_traits {
    using RefType = typename PtrTraits::ConstRefType;
    using RawPtrType = typename PtrTraits::ConstRawPtrType;
  };

  // Trait classes used to help implement symmetric operations.
  //
  // Notes about notation:
  // + LR denotes Left for the Forward version of the operation and Right
  //   for the Reverse version.
  // + RL denotes Right for the Forward version of the operation and Left
  //   for the Reverse version.
  //
  // Examples...
  // Forward : LR-child of node X == the left child of node X.
  // Reverse : LR-child of node X == the right child of node X.
  //
  // Forward : RL-most node of the tree == the right most node of the tree
  // Reverse : RL-most node of the tree == the left most node of the tree
  struct ReverseTraits;  // fwd decl
  struct ForwardTraits {
    using Inverse = ReverseTraits;

    template <typename NodeState>
    static RawPtrType& LRChild(NodeState& ns) {
      return ns.left_;
    }
    template <typename NodeState>
    static RawPtrType& RLChild(NodeState& ns) {
      return ns.right_;
    }

    static RawPtrType& LRMost(ContainerType& tree) { return tree.left_most_; }
    static RawPtrType& RLMost(ContainerType& tree) { return tree.right_most_; }
  };

  struct ReverseTraits {
    using Inverse = ForwardTraits;

    template <typename NodeState>
    static RawPtrType& LRChild(NodeState& ns) {
      return ns.right_;
    }
    template <typename NodeState>
    static RawPtrType& RLChild(NodeState& ns) {
      return ns.left_;
    }

    static RawPtrType& LRMost(ContainerType& tree) { return tree.right_most_; }
    static RawPtrType& RLMost(ContainerType& tree) { return tree.left_most_; }
  };

  // Trait classes used to define the bound condition for upper_bound and
  // lower_bound.
  struct UpperBoundTraits {
    static bool GoRight(const KeyType& key, const KeyType& node_key) {
      return KeyTraits::EqualTo(node_key, key) || KeyTraits::LessThan(node_key, key);
    }
  };

  struct LowerBoundTraits {
    static bool GoRight(const KeyType& key, const KeyType& node_key) {
      return KeyTraits::LessThan(node_key, key);
    }
  };

  // The shared implementation of the iterator
  template <class IterTraits>
  class iterator_impl {
   public:
    using value_type = ValueType;
    using reference = typename IterTraits::RefType;
    using pointer = typename IterTraits::RawPtrType;
    using difference_type = std::ptrdiff_t;
    using iterator_category = std::bidirectional_iterator_tag;

    iterator_impl() {}
    iterator_impl(const iterator_impl& other) { node_ = other.node_; }

    iterator_impl& operator=(const iterator_impl& other) {
      node_ = other.node_;
      return *this;
    }

    bool IsValid() const { return internal::valid_sentinel_ptr(node_); }
    explicit operator bool() const { return IsValid(); }
    bool operator==(const iterator_impl& other) const { return node_ == other.node_; }
    bool operator!=(const iterator_impl& other) const { return node_ != other.node_; }

    // Prefix
    iterator_impl& operator++() {
      if (IsValid())
        advance<ForwardTraits>();
      return *this;
    }

    iterator_impl& operator--() {
      // If this was a default constructed iterator, then it cannot
      // back up.
      if (node_ != nullptr) {
        // If we are at the end() of the tree, then recover the tree
        // pointer from the sentinel and back up to the right-most node.
        if (internal::is_sentinel_ptr(node_)) {
          node_ = internal::unmake_sentinel<ContainerType*>(node_)->right_most_;
        } else {
          advance<ReverseTraits>();
        }
      }

      return *this;
    }

    // Postfix
    iterator_impl operator++(int) {
      iterator_impl ret(*this);
      ++(*this);
      return ret;
    }

    iterator_impl operator--(int) {
      iterator_impl ret(*this);
      --(*this);
      return ret;
    }

    iterator_impl parent() const {
      ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node_));
      auto& ns = NodeTraits::node_state(*node_);
      return iterator_impl(ns.parent_);
    }

    iterator_impl left() const {
      ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node_));
      auto& ns = NodeTraits::node_state(*node_);
      return iterator_impl(ns.left_);
    }

    iterator_impl right() const {
      ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node_));
      auto& ns = NodeTraits::node_state(*node_);
      return iterator_impl(ns.right_);
    }

    typename PtrTraits::PtrType CopyPointer() const {
      return IsValid() ? PtrTraits::Copy(node_) : nullptr;
    }

    typename IterTraits::RefType operator*() const {
      ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node_));
      return *node_;
    }
    typename IterTraits::RawPtrType operator->() const {
      ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node_));
      return node_;
    }

   private:
    friend ContainerType;

    iterator_impl(typename PtrTraits::RawPtrType node) : node_(node) {}

    template <typename LRTraits>
    void advance() {
      ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node_));

      // Find the next node in the ordered sequence.
      // key.  This will be either...
      // 1) The RL-most child of our LR-hand sub-tree.
      // 2) Our first ancestor for which we are a LR-hand descendant.
      auto ns = &NodeTraits::node_state(*node_);
      auto rl_child = LRTraits::RLChild(*ns);
      if (rl_child != nullptr) {
        node_ = rl_child;

        // The RL-hand child of the RL-most node is terminated
        // using the sentinel value for this tree instead of nullptr.
        // Have we hit it?  If so, then we are done.
        if (internal::is_sentinel_ptr(node_))
          return;

        // While we can go LR, do so.
        auto lr_child = LRTraits::LRChild(NodeTraits::node_state(*node_));
        while (lr_child != nullptr) {
          ZX_DEBUG_ASSERT(!internal::is_sentinel_ptr(lr_child));
          node_ = lr_child;
          lr_child = LRTraits::LRChild(NodeTraits::node_state(*node_));
        }

        return;
      }

      // Climb up the tree until we traverse a LR-hand link.  Because
      // of the sentinel termination, we should never attempt to climb
      // up past the root.
      bool done;
      ns = &NodeTraits::node_state(*node_);
      do {
        ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(ns->parent_));

        auto parent_ns = &NodeTraits::node_state(*ns->parent_);
        done = (LRTraits::LRChild(*parent_ns) == node_);

        ZX_DEBUG_ASSERT(done || (LRTraits::RLChild(*parent_ns) == node_));

        node_ = ns->parent_;
        ns = parent_ns;
        } while (!done);
    }

    typename PtrTraits::RawPtrType node_ = nullptr;
  };  // class iterator_impl

  // The test framework's 'checker' class is our friend.
  friend CheckerType;

  // move semantics only
  DISALLOW_COPY_AND_ASSIGN_ALLOW_MOVE(WAVLTree);

  void internal_insert(PtrType& ptr, RawPtrType* collision = nullptr) {
    ZX_DEBUG_ASSERT(ptr != nullptr);

    auto& ns = NodeTraits::node_state(*ptr);
    ZX_DEBUG_ASSERT(ns.IsValid() && !ns.InContainer());

    // The rank of an inserted node always starts at 0.
    ns.rank_ = 0;

    // If the tree is currently empty, then this is easy.
    if (root_ == nullptr) {
      ns.parent_ = sentinel();
      ns.left_ = sentinel();
      ns.right_ = sentinel();

      ZX_DEBUG_ASSERT(internal::is_sentinel_ptr(left_most_) &&
                      internal::is_sentinel_ptr(right_most_));
      left_most_ = PtrTraits::GetRaw(ptr);
      right_most_ = PtrTraits::GetRaw(ptr);

      root_ = PtrTraits::Leak(ptr);
      ++count_;
      Observer::RecordInsert(root());
      return;
    }

    // Find the proper position for this node.
    auto key = KeyTraits::GetKey(*ptr);
    bool is_left_most = true;
    bool is_right_most = true;
    RawPtrType parent = root_;
    RawPtrType* owner;

    while (true) {
      auto parent_key = KeyTraits::GetKey(*parent);

      // Looks like we collided with an object in the collection which has
      // the same key as the object being inserted.  This is only allowed
      // during an insert_or_find operation (collision is non-null).
      // Assert this in debug builds.  If this is an insert_or_find
      // operation, fill out the collision [out] parameter so the called
      // knows which object he/she collided with.  Either way, do not
      // actually insert the object.
      if (KeyTraits::EqualTo(key, parent_key)) {
        ZX_DEBUG_ASSERT(collision && *collision == nullptr);
        *collision = parent;
        return;
      }

      // Update user-defined invariants in the subtree node only when the
      // key does not collide.
      Observer::RecordInsertTraverse(PtrTraits::GetRaw(ptr), iterator(parent));

      auto& parent_ns = NodeTraits::node_state(*parent);

      // Decide which side of the current parent-under-consideration the
      // node to be inserted belongs on.  If we are going left, then we
      // are no longer right-most, and vice-versa.
      if (KeyTraits::LessThan(key, parent_key)) {
        owner = &parent_ns.left_;
        is_right_most = false;
      } else {
        owner = &parent_ns.right_;
        is_left_most = false;
      }

      // If we would have run out of valid pointers in the direction we
      // should be searching, then we are done.
      if (!internal::valid_sentinel_ptr(*owner))
        break;

      // We belong on a side of the parent-under-consideration which
      // already has a child.  Move down to the child and consider it
      // instead.
      parent = *owner;
    }

    // We know that we are not the root of the tree, therefore we cannot be
    // both left and right-most.
    ZX_DEBUG_ASSERT(!is_left_most || !is_right_most);

    if (is_right_most) {
      ZX_DEBUG_ASSERT(internal::is_sentinel_ptr(*owner));
      ns.right_ = sentinel();
      right_most_ = PtrTraits::GetRaw(ptr);
    } else if (is_left_most) {
      ZX_DEBUG_ASSERT(internal::is_sentinel_ptr(*owner));
      ns.left_ = sentinel();
      left_most_ = PtrTraits::GetRaw(ptr);
    }

    // The owner link must either be nullptr or the sentinel.  This is
    // equivalent to saying that the pointer is not valid (using the pointer
    // traits definition of "valid").
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(*owner) == false);
    ns.parent_ = parent;

    *owner = PtrTraits::Leak(ptr);
    ++count_;

    Observer::RecordInsert(iterator(*owner));

    // Finally, perform post-insert balance operations.
    BalancePostInsert(*owner);
  }

  PtrType internal_erase(RawPtrType ptr) {
    if (!internal::valid_sentinel_ptr(ptr))
      return PtrType(nullptr);

    auto& ns = NodeTraits::node_state(*ptr);
    RawPtrType* owner = &GetLinkPtrToNode(ptr);
    ZX_DEBUG_ASSERT(*owner == ptr);

    // If the node we want to remove has two children, swap it with the
    // left-most node of the right-hand sub-tree before proceeding.  This
    // will guarantee that we are operating in a 0 or 1 child case.
    if (internal::valid_sentinel_ptr(ns.left_) && internal::valid_sentinel_ptr(ns.right_)) {
      RawPtrType* new_owner = &ns.right_;
      auto new_ns = &NodeTraits::node_state(*ns.right_);

      while (new_ns->left_ != nullptr) {
        ZX_DEBUG_ASSERT(!internal::is_sentinel_ptr(new_ns->left_));
        new_owner = &new_ns->left_;
        new_ns = &NodeTraits::node_state(*new_ns->left_);
      }

      owner = SwapWithRightDescendant(*owner, *new_owner);
      ZX_DEBUG_ASSERT(*owner == ptr);
    }

    // Now that we know our relationship with our parent, go ahead and start
    // the process of removing the node.  Keep track of the target node's
    // parent, whether it was it's parent's left or right child, and whether
    // it was a 1-child or a 2-child.  We will need this info when it comes
    // time to rebalance.
    RawPtrType parent = ns.parent_;
    bool was_one_child, was_left_child;

    ZX_DEBUG_ASSERT(parent != nullptr);
    if (!internal::is_sentinel_ptr(parent)) {
      auto& parent_ns = NodeTraits::node_state(*parent);

      was_one_child = ns.rank_parity() != parent_ns.rank_parity();
      was_left_child = &parent_ns.left_ == owner;
    } else {
      was_one_child = false;
      was_left_child = false;
    }

    // Reclaim our refernce from our owner and unlink.  We return the to the
    // caller when we are done.
    PtrType removed = PtrTraits::Reclaim(*owner);
    *owner = nullptr;

    // We know that we have at most one child.  If we do have a child,
    // promote it to our position.  While we are handling the cases,
    // maintain the LR-most bookkeeping.  Consider the 1 and 0 child cases
    // separately.
    //
    // 1-child case:
    // We are promoting the LR-child.  If we were RL-most, then our RL-child
    // is the sentinel value.  We need to transfer this sentinel value to
    // the RL-most node in our LR-subtree.  We also need to update the
    // parent pointer of our LR-child to point to the removed node's parent.
    //
    // 0-child case:
    // We are not promoting any node, but we may or may not have been the
    // LR-most.
    //
    // ++ If the target is both the left and right-most node in the tree, it
    //    is a leaf, then the target *must* be the final node in the tree.  The
    //    tree's left and right-most need to be updated to be the sentinel
    //    value, and the sentinels need to be cleared from the target node's
    //    state structure.
    // ++ If the target is the LR-most node in the tree and a leaf, then its
    //    parent is now the LR-most node in the tree.  We need to transfer
    //    the sentinel from the target's LR-child to the pointer which used
    //    to point to it, and update the LR-most bookkeeping in the tree to
    //    point to the target's parent. The target node's RL-child should
    //    already be nullptr.
    // ++ If the target neither the left nor the right most node in the
    //    tree, then nothing special needs to happen with regard to the
    //    left/right-most bookkeeping.
    RawPtrType target = PtrTraits::GetRaw(removed);
    if (internal::valid_sentinel_ptr(ns.left_)) {
      PromoteLRChild<ForwardTraits>(*owner, target);
    } else if (internal::valid_sentinel_ptr(ns.right_)) {
      PromoteLRChild<ReverseTraits>(*owner, target);
    } else {
      // The target's LR-child is the sentinel if and only if the target
      // is the LR-most node in the tree.
      ZX_DEBUG_ASSERT(internal::is_sentinel_ptr(ns.left_) == (left_most_ == target));
      ZX_DEBUG_ASSERT(internal::is_sentinel_ptr(ns.right_) == (right_most_ == target));

      if (internal::is_sentinel_ptr(ns.left_)) {
        if (internal::is_sentinel_ptr(ns.right_)) {
          // Target is both left and right most.
          ZX_DEBUG_ASSERT(count_ == 1);
          ZX_DEBUG_ASSERT(internal::is_sentinel_ptr(ns.parent_));
          left_most_ = sentinel();
          right_most_ = sentinel();
          ns.left_ = nullptr;
          ns.right_ = nullptr;
        } else {
          // Target is just left most.
          ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(ns.parent_));
          ZX_DEBUG_ASSERT(ns.right_ == nullptr);
          left_most_ = ns.parent_;
          *owner = ns.left_;
          ns.left_ = nullptr;
        }
      } else if (internal::is_sentinel_ptr(ns.right_)) {
        // Target is just right most.
        ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(ns.parent_));
        ZX_DEBUG_ASSERT(ns.left_ == nullptr);
        right_most_ = ns.parent_;
        *owner = ns.right_;
        ns.right_ = nullptr;
      }

      // Disconnect the target's parent pointer and we should be done.
      ns.parent_ = nullptr;
    }

    // At this point in time, the target node should have been completely
    // removed from the tree.  Its internal state should be valid, and
    // indicate that it is not in the container.
    ZX_DEBUG_ASSERT(ns.IsValid() && !ns.InContainer());

    // Update the count bookkeeping.
    --count_;
    Observer::RecordErase(target, iterator(parent));

    // Time to rebalance.  We know that we don't need to rebalance if we
    // just removed the root (IOW - its parent was the sentinel value).
    if (!internal::is_sentinel_ptr(parent)) {
      if (was_one_child) {
        // If the node we removed was a 1-child, then we may have just
        // turned its parent into a 2,2 leaf node.  If so, we have a
        // leaf node with non-zero rank and need to fix the problem.
        BalancePostErase_Fix22Leaf(parent);
      } else {
        // If the node we removed was a 2-child, the we may have just
        // created a 3 child which we need to fix.  The balance routine
        // will handle checking for us, but it will need to know whether
        // the node we removed was its parent's left or right child in
        // order to un-ambiguously perform the check.
        if (was_left_child)
          BalancePostErase_FixLR3Child<ForwardTraits>(parent);
        else
          BalancePostErase_FixLR3Child<ReverseTraits>(parent);
      }
    }

    // Give the reference to the node we just removed back to the caller.
    return removed;
  }

  // internal::Swap a node which is currently in the tree with a new node.
  //
  // old_node *must* already be in the tree.
  // new_node *must* not be in the tree.
  // old_node and new_node *must* have the same key.
  //
  PtrType internal_swap(RawPtrType old_node, PtrType&& new_node) {
    ZX_DEBUG_ASSERT(old_node != nullptr);
    ZX_DEBUG_ASSERT(new_node != nullptr);
    ZX_DEBUG_ASSERT(KeyTraits::EqualTo(KeyTraits::GetKey(*old_node), KeyTraits::GetKey(*new_node)));

    auto& old_ns = NodeTraits::node_state(*old_node);
    auto& new_ns = NodeTraits::node_state(*new_node);
    auto new_raw = PtrTraits::GetRaw(new_node);

    ZX_DEBUG_ASSERT(old_ns.InContainer());
    ZX_DEBUG_ASSERT(!new_ns.InContainer());

    // Start with the left child state.
    if (internal::valid_sentinel_ptr(old_ns.left_)) {
      // Fix the left-child's parent pointer.
      NodeTraits::node_state(*old_ns.left_).parent_ = new_raw;
    } else {
      // We have no left child, so there is no left-child parent pointer
      // to fixup, but we may need to fix the left-most bookkeeping.
      if (internal::is_sentinel_ptr(old_ns.left_)) {
        ZX_DEBUG_ASSERT(left_most_ == old_node);
        left_most_ = new_raw;
      }
    }
    new_ns.left_ = old_ns.left_;
    old_ns.left_ = nullptr;

    // Same routine, but this time with the right child state.
    if (internal::valid_sentinel_ptr(old_ns.right_)) {
      // Fix the right-child's parent pointer.
      NodeTraits::node_state(*old_ns.right_).parent_ = new_raw;
    } else {
      // We have no right child, so there is no right-child parent pointer
      // to fixup, but we may need to fix the right-most bookkeeping.
      if (internal::is_sentinel_ptr(old_ns.right_)) {
        ZX_DEBUG_ASSERT(right_most_ == old_node);
        right_most_ = new_raw;
      }
    }
    new_ns.right_ = old_ns.right_;
    old_ns.right_ = nullptr;

    // Don't forget transfer the rank bookkeeping from the old node to the
    // new node.
    new_ns.rank_ = old_ns.rank_;

    // Finally, update the pointer which pointed to the old node to point to
    // the new node, taking ownership of the managed reference (if any) in
    // the process.  The update the parent pointers, and finally reclaim the
    // reference from the node we just swapped out and give it back to the
    // caller.
    GetLinkPtrToNode(old_node) = PtrTraits::Leak(new_node);
    new_ns.parent_ = old_ns.parent_;
    old_ns.parent_ = nullptr;
    return PtrTraits::Reclaim(old_node);
  }

  template <typename BoundTraits>
  const_iterator internal_upper_lower_bound(const KeyType& key) const {
    RawPtrType node = root_;
    RawPtrType found = sentinel();

    while (internal::valid_sentinel_ptr(node)) {
      auto node_key = KeyTraits::GetKey(*node);

      if (BoundTraits::GoRight(key, node_key)) {
        // If we need to look for a larger node value (key > node_key in
        // the case of lower_bound, key >= node_key in the case of
        // upper_bound), then go right.  If we cannot go right, then
        // there is other element in this container whose key satisfies
        // the bound condition.  Break out of the loop and return the
        // best node we have found so far.
        auto& ns = NodeTraits::node_state(*node);

        if (ns.right_ == nullptr)
          break;

        node = ns.right_;
      } else {
        // If this node's key must be greater than or equal to the
        // user's key.  This node is now our candidate for our found
        // node.
        found = node;

        // If this node has a left-hand child, it is possible that there
        // is a better bound somewhere underneath it.  Set the node
        // pointer to the root of the left hand sub-tree and keep
        // looking.
        node = NodeTraits::node_state(*node).left_;
      }
    }

    return const_iterator(found);
  }

  constexpr RawPtrType sentinel() const { return internal::make_sentinel<RawPtrType>(this); }

  // Swaps the positions of two nodes, one of which is guaranteed to be a
  // right-hand descendant of the other.
  //
  // @note Node #1 is the ancestor node while node #2 is the descendant node.
  //
  // @param  ptr_ref1 A reference to the pointer which points to node #1.
  // @param  ptr_ref2 A reference to the pointer which points to node #2.
  // @return The new pointer to the pointer which points to node #1.
  //
  RawPtrType* SwapWithRightDescendant(RawPtrType& ptr_ref1, RawPtrType& ptr_ref2) {
    RawPtrType node1 = ptr_ref1;
    RawPtrType node2 = ptr_ref2;

    auto& ns1 = NodeTraits::node_state(*node1);
    auto& ns2 = NodeTraits::node_state(*node2);

    auto ns1_lp = internal::valid_sentinel_ptr(ns1.left_)
                      ? &NodeTraits::node_state(*ns1.left_).parent_
                      : nullptr;
    auto ns2_lp = internal::valid_sentinel_ptr(ns2.left_)
                      ? &NodeTraits::node_state(*ns2.left_).parent_
                      : nullptr;
    auto ns2_rp = internal::valid_sentinel_ptr(ns2.right_)
                      ? &NodeTraits::node_state(*ns2.right_).parent_
                      : nullptr;

    // node 2 is a right-hand descendant of node 1, so node 1's right hand
    // pointer must be valid.
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(ns1.right_));
    auto ns1_rp = &NodeTraits::node_state(*ns1.right_).parent_;

    // Start by updating the LR-most bookkeeping.  Node 1 cannot be the
    // right-most node, and node 2 cannot be the left most node (because we
    // know that node 2 is to the right of node 1).
    //
    // If node 1 is currently the left-most, then node 2 will be when we are
    // done.  Likewise, if node 2 is currently the right-most, then node 1
    // will be the right-most when we are done.
    if (node1 == left_most_)
      left_most_ = node2;
    if (node2 == right_most_)
      right_most_ = node1;

    // Next, swap the core state of node 1 and node 2.
    internal::Swap(ns1.parent_, ns2.parent_);
    internal::Swap(ns1.left_, ns2.left_);
    internal::Swap(ns1.right_, ns2.right_);
    internal::Swap(ns1.rank_, ns2.rank_);

    // At this point, there are two scenarios.
    //
    // Case #1
    // Node 2 was an indirect descendant of node 1.  In this case, all we
    // need to do is swap the ptr_ref[12] pointers and fix-up the various
    // children's parent pointers (when they exist).
    //
    // Case #2
    // Node 2 was the direct descendant of node 1; in this case the right
    // hand child.  In this case, we know 2 things...
    //
    // 1) node1.right is the same pointer as ptr_ref2
    // 2) node2.parent is the same pointer as node 1's right-child's parent.
    //
    // Because of this (and because of the swapping of core state prior to
    // this), we know...
    //
    // 1) node1.parent currently points to node 1, but should point to node 2.
    // 2) node2.right currently points to node 2, but should point to node 1.
    // 3) ptr_ref1 still points to node 1, but should point to node 2.
    // 4) node2.parent (aka; ns1_rp) currently points to node 1's old
    //    parent (which is correct).
    //
    // We fix issues 1-3 by swapping ptr_ref1 and node2.right, and by
    // directly setting setting node1.parent to node2.
    //
    // Finally, we need to return the new pointer to the pointer which
    // points to node 1.  In case #1, this is just ptr_ref2.  In case #2,
    // however, this has become node 2's right hand pointer.
    //
    // Perform the common child fixup first, then deal with the special
    // cases.
    if (ns1_lp)
      *ns1_lp = node2;
    if (ns2_lp)
      *ns2_lp = node1;
    if (ns2_rp)
      *ns2_rp = node1;

    if (&ptr_ref2 != &ns1.right_) {
      // Case #1.
      internal::Swap(ptr_ref1, ptr_ref2);
      *ns1_rp = node2;
      return &ptr_ref2;
    } else {
      ZX_DEBUG_ASSERT(ns1.parent_ == node1);
      ZX_DEBUG_ASSERT(ns2.right_ == node2);
      internal::Swap(ptr_ref1, ns2.right_);
      ns1.parent_ = node2;
      return &ns2.right_;
    }
  }

  // Promote the LR-child of a node to be removed into the node's position.
  // Update the LR-node bookkeeping in the process.  The LRTraits will
  // determine if we are promoting the left or right child (the forward
  // operation promotes left).
  //
  // Requirements:
  // ++ The node being removed *must* have an LR-child.
  // ++ The node being removed *must not* have an RL-child.
  // ++ The owner reference *must* have already been disconnected from the
  //    node to be removed.
  //
  // @param owner A reference to the pointer which points to the node to be
  // removed.
  // @param node A pointer to the node to be removed.
  //
  template <typename LRTraits>
  void PromoteLRChild(RawPtrType& owner, RawPtrType node) {
    ZX_DEBUG_ASSERT(owner == nullptr);
    ZX_DEBUG_ASSERT(node != nullptr);

    auto& ns = NodeTraits::node_state(*node);
    RawPtrType& lr_child = LRTraits::LRChild(ns);
    RawPtrType& rl_child = LRTraits::RLChild(ns);

    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(lr_child) &&
                    !internal::valid_sentinel_ptr(rl_child));

    // Promote by transferring the LR-Child pointer to the owner pointer and
    // fixing up the LR-Child's parent pointer be the current parent of the
    // node to be removed.
    owner = lr_child;    // owner now points to the promoted node.
    lr_child = nullptr;  // the node being removed no longer has any lr child.
    NodeTraits::node_state(*owner).parent_ = ns.parent_;

    // The removed node is the RL-most node if (and only if) its RL-child
    // was the sentinel value.
    RawPtrType& rl_most = LRTraits::RLMost(*this);
    ZX_DEBUG_ASSERT((rl_most == node) == internal::is_sentinel_ptr(rl_child));
    if (internal::is_sentinel_ptr(rl_child)) {
      // The target node was the RL-most.  Find the new RL-most node. It will
      // be the RL-most node in the LR-subtree of the target node.  Once
      // found, update the RL-child of the new RL-most node to be the
      // sentinel value.
      RawPtrType replacement = owner;
      RawPtrType* next_rl_child;

      while (true) {
        auto& replacement_ns = NodeTraits::node_state(*replacement);
        next_rl_child = &LRTraits::RLChild(replacement_ns);

        ZX_DEBUG_ASSERT(!internal::is_sentinel_ptr(*next_rl_child));
        if (*next_rl_child == nullptr)
          break;

        replacement = *next_rl_child;
      }

      // Update the bookkeeping.
      rl_most = replacement;
      *next_rl_child = sentinel();
      rl_child = nullptr;
    }

    // Unlink the parent pointer for the target node and we should be done.
    // The left and right children of the target node should already be
    // nullptr by now.
    ns.parent_ = nullptr;
    ZX_DEBUG_ASSERT(ns.left_ == nullptr);
    ZX_DEBUG_ASSERT(ns.right_ == nullptr);
  }

  // After we have swapped contents with another tree, we need to fix up the
  // sentinel values so that they refer to the proper tree.  Otherwise tree
  // A's sentinels will point at tree B's, and vice-versa.
  void FixSentinelsAfterSwap() {
    if (root_) {
      ZX_DEBUG_ASSERT(!internal::is_sentinel_ptr(root_));
      ZX_DEBUG_ASSERT(left_most_ && !internal::is_sentinel_ptr(left_most_));
      ZX_DEBUG_ASSERT(right_most_ && !internal::is_sentinel_ptr(right_most_));

      auto& root_ns = NodeTraits::node_state(*root_);
      auto& left_most_ns = NodeTraits::node_state(*left_most_);
      auto& right_most_ns = NodeTraits::node_state(*right_most_);

      ZX_DEBUG_ASSERT(internal::is_sentinel_ptr(left_most_ns.left_));
      ZX_DEBUG_ASSERT(internal::is_sentinel_ptr(right_most_ns.right_));

      root_ns.parent_ = sentinel();
      left_most_ns.left_ = sentinel();
      right_most_ns.right_ = sentinel();
    } else {
      ZX_DEBUG_ASSERT(internal::is_sentinel_ptr(left_most_));
      ZX_DEBUG_ASSERT(internal::is_sentinel_ptr(right_most_));
      left_most_ = sentinel();
      right_most_ = sentinel();
    }
  }

  // GetLinkPtrToNode.
  //
  // Obtain a reference to the pointer which points to node.  The will either be
  // a reference to the node's parent's left child, right child, or the root
  // node of the tree if the child has no parent.
  RawPtrType& GetLinkPtrToNode(RawPtrType node) {
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node));

    auto& ns = NodeTraits::node_state(*node);
    if (internal::is_sentinel_ptr(ns.parent_)) {
      ZX_DEBUG_ASSERT(ns.parent_ == sentinel());
      ZX_DEBUG_ASSERT(root_ == node);
      return root_;
    }

    ZX_DEBUG_ASSERT(ns.parent_ != nullptr);
    auto& parent_ns = NodeTraits::node_state(*ns.parent_);
    if (parent_ns.left_ == node)
      return parent_ns.left_;

    ZX_DEBUG_ASSERT(parent_ns.right_ == node);
    return parent_ns.right_;
  }

  // RotateLR<LRTraits>
  //
  // Perform a Rotate-LR operation at 'node'.
  //
  // Let...
  // X = node
  // Y = LR-Child of X (if any)
  // Z = X's parent.
  //
  // A Rotate-LR operation at 'node' is defined as follows.
  // 1) Z becomes the LR-Child of X
  // 2) Y becomes the RL-Child of Z
  // 3) X takes Z's position in the tree
  //
  // If [XYZ]_link is the pointer which points to [XYZ], then we can describe
  // the operation as the following permutation of the links.
  //
  //   link  | before | swap1 | swap2 |
  // --------+--------+-------+-------+
  //  X_link |   X    |   Y   |   Y   +
  //  Y_link |   Y    |   X   |   Z   +
  //  Z_link |   Z    |   Z   |   X   +
  //
  // Note that link's are bi-directional.  We need to update the parent
  // pointers as well.  Let G be Z's parent at the start of the operation.
  // The permutation should be.
  //
  //   node  | P(n) before | P(n) after |
  // --------+-------------+------------+
  //    X    |      Z      |     G      |
  //    Y    |      X      |     Z      |
  //    Z    |      G      |     X      |
  //
  // We should not need to worry about the LR-most-ness of any of the nodes.
  // Reasoning is as follows...
  //
  // 1) Z might be the LR-most node in the tree, but only if it has no
  //    LR-Child.  It cannot be the RL-most node in the tree, because we know
  //    that it has an RL-child (X).  When we rotate, Z moves to the LR.  So
  //    if it or one of its LR-children was LR-most, they remain that way.
  // 2) X might be the RL-most node in the tree, but only if it has no
  //    RL-child.  It cannot be the LR-most node in the tree because it is the
  //    RL-child of Z.  When we rotate, X moves up in the tree following Z's
  //    RL-child link.  This means that if X was on the RL-edge of the tree
  //    (implying that it or one of its children was RL-most), it remains
  //    there, preserving the RL-most-ness of it or its children.
  // 3) Before the rotation, Y cannot be either the RL-most node in the tree
  //    (it is X's LR-child), or the LR-most node in the tree (it's parent, X,
  //    is Z's RL-child).  After the rotation, Y still cannot be either
  //    LR-most (it is now Z's RL-child) or RL-most (it's parent, Z, is X's
  //    LR-child).
  template <typename LRTraits>
  void RotateLR(RawPtrType node, RawPtrType parent) {
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node));    // Node must be valid
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(parent));  // Node must have a parent

    // Aliases, just to make the code below match the notation used above.
    RawPtrType X = node;
    RawPtrType Z = parent;

    auto& X_ns = NodeTraits::node_state(*X);
    auto& Z_ns = NodeTraits::node_state(*Z);

    // X must be the RL-child of Z.
    ZX_DEBUG_ASSERT(LRTraits::RLChild(Z_ns) == X);

    RawPtrType& X_link = LRTraits::RLChild(Z_ns);
    RawPtrType& Y_link = LRTraits::LRChild(X_ns);
    RawPtrType& Z_link = GetLinkPtrToNode(Z);

    RawPtrType G = Z_ns.parent_;
    RawPtrType Y = Y_link;

    // The pointer to Y cannot be a sentinel, because that would imply that
    // X was LR-most.
    ZX_DEBUG_ASSERT(!internal::is_sentinel_ptr(Y));

    // Record the rotation observation before the links are updated. The children are specified here
    // as a left rotation. These are transformed by the LRTraits for the right rotation case.
    Observer::RecordRotation(iterator(X), iterator(Y), iterator(LRTraits::RLChild(X_ns)),
                             iterator(Z), iterator(LRTraits::LRChild(Z_ns)));

    // Permute the downstream links.
    RawPtrType tmp = X_link;
    X_link = Y_link;
    Y_link = Z_link;
    Z_link = tmp;

    // Update the parent pointers (note that Y may not exist).
    X_ns.parent_ = G;
    Z_ns.parent_ = X;
    if (Y) {
      NodeTraits::node_state(*Y).parent_ = Z;
    }
  }

  // PostInsertFixupLR<LRTraits>
  //
  // Called after the promotion stage of an insert operation, when node's
  // parent has become 0,2 node and the rank rule needs to be restored.
  //
  // If node is the LR-child of parent, define...
  // X = node
  // Y = RL-child of X
  // Z = parent
  //
  // There are two possibilities of what to do next.
  //
  // if (Y is null) or (Y is a 2-child)
  // then...
  // 1) rotate-RL about X
  // 2) demote Z
  //
  // OR
  //
  // if (Y is a 1-child)
  // then...
  // 1) rotate-LR about Y
  // 2) rotate-RL about Y
  // 3) promote Y
  // 4) demote X
  // 5) demote Z
  template <typename LRTraits>
  void PostInsertFixupLR(RawPtrType node, RawPtrType parent) {
    using RLTraits = typename LRTraits::Inverse;

    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node));
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(parent));

    auto& node_ns = NodeTraits::node_state(*node);
    auto& parent_ns = NodeTraits::node_state(*parent);

    ZX_DEBUG_ASSERT(LRTraits::LRChild(parent_ns) == node);

    RawPtrType rl_child = LRTraits::RLChild(node_ns);
    auto rl_child_ns =
        internal::valid_sentinel_ptr(rl_child) ? &NodeTraits::node_state(*rl_child) : nullptr;
    if (!rl_child_ns || (rl_child_ns->rank_parity() == node_ns.rank_parity())) {
      // Case #1; single rotation
      //
      // Start by performing a Rotate-RL at node
      RotateLR<RLTraits>(node, parent);
      Observer::RecordInsertRotation();

      // Now demote node's old parent (now its LR-child)
      parent_ns.demote_rank();
    } else {
      // Case #2; double rotation.
      //
      // Start by performing a Rotate-LR at rl_child.  Afterwards,
      // rl_child will be LR-child of parent (and node is now its
      // LR-child).  Rotate-RL at rl_child.
      RotateLR<LRTraits>(rl_child, node);
      RotateLR<RLTraits>(rl_child, parent);
      Observer::RecordInsertDoubleRotation();

      // 1 promotion and 2 demotions.
      rl_child_ns->promote_rank();
      node_ns.demote_rank();
      parent_ns.demote_rank();
    }
  }

  // BalancePostInsert
  //
  // Execute the bottom-up post-insert rebalancing algorithm.
  //
  // @param node A pointer to the node which was just inserted.
  //
  void BalancePostInsert(RawPtrType node) {
    // We do not balance the tree after inserting the first (root)
    // node, so we should be able to assert that we have a valid parent.
    auto node_ns = &NodeTraits::node_state(*node);
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node_ns->parent_));

    // If we have a sibling, then our parent just went from being a 1,2
    // unary node into a 1,1 binary node and no action needs to be taken.
    RawPtrType parent = node_ns->parent_;
    auto parent_ns = &NodeTraits::node_state(*parent);
    if (internal::valid_sentinel_ptr(parent_ns->left_) &&
        internal::valid_sentinel_ptr(parent_ns->right_))
      return;

    // We have no sibling, therefor we just changed our parent from a 1,1
    // leaf node into a 0,1 unary node.  The WAVL rank rule states that all
    // rank differences are 1 or 2; 0 is not allowed.  Rebalance the tree in
    // order to restore the rule.
    //
    // Start with the promotions.  Pseudo code is...
    //
    // while (IsValid(parent) && parent is a 0,1 node)
    //    promote parent
    //    climb up the tree
    bool node_parity;
    bool parent_parity;
    bool sibling_parity;
    bool is_left_child;
    do {
      // Promote
      parent_ns->promote_rank();
      Observer::RecordInsertPromote();

      // Climb
      node = parent;
      node_ns = &NodeTraits::node_state(*node);
      parent = node_ns->parent_;

      // If we have run out of room to climb, then we must be done.
      if (!internal::valid_sentinel_ptr(parent))
        return;

      // We have a parent.  Determine the relationship between our rank
      // parity,  our parent's rank parity, and our sibling's rank parity
      // (if any).  Note; null children have a rank of -1, therefor the
      // rank parity of a sibling is always odd (1).  In the process, make
      // a note of whether we are our parent's left or right child.
      parent_ns = &NodeTraits::node_state(*parent);
      is_left_child = (parent_ns->left_ == node);
      if (is_left_child) {
        sibling_parity = internal::valid_sentinel_ptr(parent_ns->right_)
                             ? NodeTraits::node_state(*parent_ns->right_).rank_parity()
                             : true;
      } else {
        ZX_DEBUG_ASSERT(parent_ns->right_ == node);
        sibling_parity = internal::valid_sentinel_ptr(parent_ns->left_)
                             ? NodeTraits::node_state(*parent_ns->left_).rank_parity()
                             : true;
      }

      node_parity = node_ns->rank_parity();
      parent_parity = parent_ns->rank_parity();

      // We need to keep promoting and climbing if we just turned our new
      // parent into a 0,1 node.  Let N, P and S denote the current node,
      // parent, and sibling parities.  Working out the truth tables, our
      // parent is now a 0,1 node iff (!N * !P * S) + (N * P * !S)
    } while ((!node_parity && !parent_parity && sibling_parity) ||
             (node_parity && parent_parity && !sibling_parity));

    // OK.  At this point, we know that we have a parent, and our parent is
    // not a 0,1 node.  Either our rank rule has been restored and we are
    // done, or our parent is 0,2 and we need to perform either one or two
    // rotations.  Start by checking to see if our parent is 0,2.  Using the
    // notation from above...
    //
    // ++ our parent is a 0,2 node iff (!N * !P * !S) + (N * P * S).
    // ++ which implies that we are finished iff (N != P) + (N != S)
    //
    if ((node_parity != parent_parity) || (node_parity != sibling_parity))
      return;

    // Looks like our parent is a 0,2 node.  Perform the post-insert fixup
    // operations, forward if we are our parent's left child, reverse if we
    // are our parent's right child.
    if (is_left_child)
      PostInsertFixupLR<ForwardTraits>(node, parent);
    else
      PostInsertFixupLR<ReverseTraits>(node, parent);
  }

  // BalancePostErase_Fix22Leaf
  //
  // Called after an erase operation which erased the 1-child of a node.
  // Checks to see if the node has become a 2,2 leaf node and takes
  // appropriate action to restore the rank rule if needed.
  void BalancePostErase_Fix22Leaf(RawPtrType node) {
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node));

    // If we just turned node into a 2,2 leaf, it will have no children and
    // odd rank-parity.  If it has even parity, or any children at all,
    // there is nothing we need to do.
    auto& ns = NodeTraits::node_state(*node);
    if (!ns.rank_parity() || internal::valid_sentinel_ptr(ns.left_) ||
        internal::valid_sentinel_ptr(ns.right_))
      return;

    // Demote the node turning it into a 1,1 leaf.
    ns.demote_rank();
    Observer::RecordEraseDemote();

    // By demoting this node, we may have just created a 3-child.  Find its
    // parent, figure out if it was the left or right child, then use the
    // FixLR3Child method to check for the 3-child case and deal with it if
    // we need to.  If this node had no parent, then we know that we are
    // finished.
    ZX_DEBUG_ASSERT(ns.parent_ != nullptr);
    if (internal::is_sentinel_ptr(ns.parent_))
      return;

    auto& parent_ns = NodeTraits::node_state(*ns.parent_);
    bool is_left_child = parent_ns.left_ == node;
    ZX_DEBUG_ASSERT(is_left_child || (parent_ns.right_ == node));

    if (is_left_child)
      BalancePostErase_FixLR3Child<ForwardTraits>(ns.parent_);
    else
      BalancePostErase_FixLR3Child<ReverseTraits>(ns.parent_);
  }

  // BalancePostErase_FixLR3Child<LRTraits>
  //
  // Called during post-erase rebalancing when it is possible that a 3-child
  // may have been created.  There are 3 ways that this may have happened.
  //
  // 1) A node's 2-child leaf node was erased making the link to null a
  //    3-child link.
  // 2) A node's 2-child unary node was erased making the promoted node
  //    a 3-child.
  // 3) During rebalancing fix-up of a 2,2 leaf, the 2,2 leaf node is a
  //    2-child.  Demoting it to turn it into a 1,1 leaf makes it a 3-child.
  //
  // Note: this method is templated using LRTraits because the method needs to
  // know which link may have become a 3-child based on prior circumstances.
  // Without this knowledge, it is not possible to detect a 3 child using only
  // rank parity.  For this method, the LR-Child of 'node' is known to now be
  // either a 2-child or a 3-child, it cannot be a 1-child.
  template <typename LRTraits>
  void BalancePostErase_FixLR3Child(RawPtrType node) {
    using RLTraits = typename LRTraits::Inverse;
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(node));

    // Throughout this method, we will use the following notation.
    //
    // Let...
    // Z = The node with the (potential) 3-child.
    // X = The (potential) 3-child.
    // Y = X's sibling. (if any)
    //
    RawPtrType Z = node;
    auto Z_ns = &NodeTraits::node_state(*Z);
    RawPtrType X = LRTraits::LRChild(*Z_ns);

    // Check to see if X is a 3-child of Z.  We know it is if...
    // 1) X exists and Z has odd rank parity
    // 2) X does not exist and Z has even rank parity
    if (internal::valid_sentinel_ptr(X) != Z_ns->rank_parity())
      return;

    // Phase 1, demotions.
    //
    // While X is a 3-child and Y is a 2-child, or a 2,2 node
    //    Demote Y if it is is a 2,2 node
    //    Demote Z
    //    Climb (set Z = Z-parent, update definitions of X and Y)
    //
    bool X_is_LR_child = true;
    RawPtrType Y = LRTraits::RLChild(*Z_ns);
    while (true) {
      // We know that X is a 3 child, Determine the status of Y.  We know
      // that whenever X is a 3-child that Y must exist because...
      //
      // 1) Z's rank is at least 2. In the case that X does not exist, X's rank
      //    is -1 so Z's rank is (-1 + 3) = 2.  If X does exist, Z's rank
      //    is even larger.
      // 2) The rank rule for the Y,Z relationship should currently hold,
      //    meaning that the rank difference between Y and Z is either 1 or 2,
      //    therefor Y's rank is at least 0.
      // 3) Because Y has non-negative rank, it must exist.
      ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(Y));

      auto& Y_ns = NodeTraits::node_state(*Y);
      bool Y_is_2_child = (Y_ns.rank_parity() == Z_ns->rank_parity());

      // Our next steps are already determined and don't involve Y if
      // Y is currently a 2 child.  Don't bother to compute
      // Y_is_22_node if we already know that Y is a 2-child.
      if (!Y_is_2_child) {
        bool Y_is_22_node;
        if (Y_ns.rank_parity()) {
          // If Y has odd rank parity, it is a 2,2 node if both its
          // children have odd parity, meaning each child either does
          // not exist, or exists and has odd parity..
          Y_is_22_node = ((!internal::valid_sentinel_ptr(Y_ns.left_) ||
                           NodeTraits::node_state(*Y_ns.left_).rank_parity()) &&
                          (!internal::valid_sentinel_ptr(Y_ns.right_) ||
                           NodeTraits::node_state(*Y_ns.right_).rank_parity()));
        } else {
          // If Y has even rank parity, it can only be a 2,2 node if it is
          // a binary node and both of its children have even parity.
          Y_is_22_node = internal::valid_sentinel_ptr(Y_ns.left_) &&
                         internal::valid_sentinel_ptr(Y_ns.right_) &&
                         !NodeTraits::node_state(*Y_ns.left_).rank_parity() &&
                         !NodeTraits::node_state(*Y_ns.right_).rank_parity();
        }

        // If Y is neither a 2-child or a 2,2 node, then we are done
        // with phase 1.  X is still a 3-child, but it will take one or
        // more rotations to fix the problem.
        if (!Y_is_22_node)
          break;
      }

      // Demote Z.  If Y was a 1-child, demote Y as well.
      Z_ns->demote_rank();
      Observer::RecordEraseDemote();

      if (!Y_is_2_child) {
        Y_ns.demote_rank();
        Observer::RecordEraseDemote();
      }

      // Climb.  If we cannot climb because we have no parent, or we can
      // climb and the new X is no longer a 3-child, then we are done
      // with the rebalance operation.
      if (!internal::valid_sentinel_ptr(Z_ns->parent_))
        return;

      bool X_rank_parity = Z_ns->rank_parity();
      X = Z;
      Z = Z_ns->parent_;
      Z_ns = &NodeTraits::node_state(*Z);
      if (Z_ns->rank_parity() == X_rank_parity)
        return;

      X_is_LR_child = (LRTraits::LRChild(*Z_ns) == X);
      Y = X_is_LR_child ? LRTraits::RLChild(*Z_ns) : LRTraits::LRChild(*Z_ns);
    }

    // Phase 2, rotations
    //
    // At this point, we know...
    //
    // 1) Z exists
    // 2) X is a 3-child of Z (but may not exist).
    // 3) Y exists (because X is a 3-child of Z, see above)
    // 4) Y is not a 2-child of Z (so, Z is a 1,3 node)
    // 5) Y is not a 2,2 node.
    //
    // We will need to perform either 1 or 2 rotations to fix this problem.
    // Which directions we rotate will now depend on whether or not X is a
    // left or right child of Z.  Invoke the post-erase rotation method
    // using the traits which will normalize the notation so that X is an
    // LR-child of Z.
    if (X_is_LR_child)
      BalancePostErase_DoRotations<LRTraits>(Y, Z);
    else
      BalancePostErase_DoRotations<RLTraits>(Y, Z);
  }

  // BalancePostErase_DoRotations<LRTraits>
  //
  // Refer to the notes at the end of BalancePostErase_FixLR3Child<LRTraits>
  // for a description of the current situation.  In addition to the
  // assertions there, we also now know that X is the LR-Child of Z (and
  // therefor Y is the RL-child of Z) (note; X is only indirectly involved
  // here, so not passed as an argument).
  template <typename LRTraits>
  void BalancePostErase_DoRotations(RawPtrType Y, RawPtrType Z) {
    using RLTraits = typename LRTraits::Inverse;
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(Y));
    ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(Z));

    auto& Y_ns = NodeTraits::node_state(*Y);
    auto& Z_ns = NodeTraits::node_state(*Z);

    // Let...
    // V = the LR-child of Y
    // W = the RL-child of Y
    //
    // Perform rotations to fix the fact that X is a 3-child of Z.  We will
    // need to do 1 of 2 things depending on whether W is a 1-child or
    // 2-child of Y.
    //
    RawPtrType W = LRTraits::RLChild(Y_ns);
    bool W_rank_parity =
        internal::valid_sentinel_ptr(W) ? NodeTraits::node_state(*W).rank_parity() : true;
    if (Y_ns.rank_parity() != W_rank_parity) {
      // Case #1: W is a 1-child of Y
      //
      // 1)  Rotate-LR at Y
      // 2)  Promote Y
      // 3a) If Z is a leaf, demote it twice.
      // 3b) else demote Z once.
      RotateLR<LRTraits>(Y, Z);
      Observer::RecordEraseRotation();

      Y_ns.promote_rank();

      if (!internal::valid_sentinel_ptr(Z_ns.left_) && !internal::valid_sentinel_ptr(Z_ns.right_)) {
        Z_ns.double_demote_rank();
      } else {
        Z_ns.demote_rank();
      }
    } else {
      // Case #2: W is a 2-child of Y
      //
      // 1) Rotate-RL at V
      // 2) Rotate-LR at V
      // 3) Promote V twice.
      // 3) Demote Y once.
      // 3) Demote Z twice.
      RawPtrType V = LRTraits::LRChild(Y_ns);
      ZX_DEBUG_ASSERT(internal::valid_sentinel_ptr(V));  // V must exist
      auto& V_ns = NodeTraits::node_state(*V);
      ZX_DEBUG_ASSERT(V_ns.rank_parity() != Y_ns.rank_parity());  // V must be a 1-child of Y

      // TODO(johngro) : Special case the implementation of a double
      // rotation operation.  It would almost certainly be more efficient
      // than performing 2 sequential single rotation operations.
      RotateLR<RLTraits>(V, Y);
      RotateLR<LRTraits>(V, Z);
      Observer::RecordEraseDoubleRotation();

      V_ns.double_promote_rank();
      Y_ns.demote_rank();
      Z_ns.double_demote_rank();
    }
  }

  // Tree state consists of...
  //
  // 1) a root node
  // 2) a left-most node
  // 3) a right-most node
  // 4) a count of nodes
  //
  // Technically, only #1 is required.  #2-4 are optimizations to assist in
  // iteration and size operations.
  RawPtrType root_ = nullptr;
  RawPtrType left_most_ = sentinel();
  RawPtrType right_most_ = sentinel();
  size_t count_ = 0;
};

// TaggedWAVLTree<> is intended for use with ContainableBaseClasses<>.
//
// For an easy way to allow instances of your class to live in multiple
// intrusive containers at once, simply derive from
// ContainableBaseClasses<YourContainables<PtrType, TagType>...> and then use
// this template instead of WAVLTree<> as the container, passing the same tag
// type you used earlier as the third parameter.
//
// See comments on ContainableBaseClasses<> in fbl/intrusive_container_utils.h
// for more details.
//
template <typename KeyType, typename PtrType, typename TagType,
          typename KeyTraits = DefaultKeyedObjectTraits<
              KeyType, typename internal::ContainerPtrTraits<PtrType>::ValueType>,
          typename Observer = tests::intrusive_containers::DefaultWAVLTreeObserver>
using TaggedWAVLTree = WAVLTree<KeyType, PtrType, KeyTraits, TagType,
                                DefaultWAVLTreeTraits<PtrType, TagType>, Observer>;

template <typename PtrType, typename TagType, NodeOptions Options = NodeOptions::None>
using TaggedWAVLTreeContainable = WAVLTreeContainable<PtrType, Options, TagType>;

}  // namespace fbl

#endif  // FBL_INTRUSIVE_WAVL_TREE_H_