static node_ptr begin_node(const_node_ptr header) noexcept;

Requires: 'header' is the header node of a tree.

Effects: Returns the first node of the tree, the header if the tree is empty.

Complexity: Constant time.

Throws: Nothing.

static node_ptr end_node(const_node_ptr header) noexcept;

Requires: 'header' is the header node of a tree.

Effects: Returns the header of the tree.

Complexity: Constant time.

Throws: Nothing.

static node_ptr root_node(const_node_ptr header) noexcept;

Requires: 'header' is the header node of a tree.

Effects: Returns the root of the tree if any, header otherwise

Complexity: Constant time.

Throws: Nothing.

static bool unique(const_node_ptr n) noexcept;

Requires: 'n' is a node of the tree or a node initialized by init(...) or init_node.

Effects: Returns true if the node is initialized by init() or init_node().

Complexity: Constant time.

Throws: Nothing.

static node_ptr get_header(const_node_ptr n);

Requires: 'n' is a node of the tree or a header node.

Effects: Returns the header of the tree.

Complexity: Logarithmic.

Throws: Nothing.

static void swap_nodes(node_ptr node1, node_ptr node2) noexcept;

Requires: node1 and node2 can't be header nodes of two trees.

Effects: Swaps two nodes. After the function node1 will be inserted in the position node2 before the function. node2 will be inserted in the position node1 had before the function.

Complexity: Logarithmic.

Throws: Nothing.

Note: This function will break container ordering invariants if node1 and node2 are not equivalent according to the ordering rules.

Experimental function

static void swap_nodes(node_ptr node1, node_ptr header1, node_ptr node2, 
                       node_ptr header2) noexcept;

Requires: node1 and node2 can't be header nodes of two trees with header header1 and header2.

Effects: Swaps two nodes. After the function node1 will be inserted in the position node2 before the function. node2 will be inserted in the position node1 had before the function.

Complexity: Constant.

Throws: Nothing.

Note: This function will break container ordering invariants if node1 and node2 are not equivalent according to the ordering rules.

Experimental function

static void replace_node(node_ptr node_to_be_replaced, node_ptr new_node) noexcept;

Requires: node_to_be_replaced must be inserted in a tree and new_node must not be inserted in a tree.

Effects: Replaces node_to_be_replaced in its position in the tree with new_node. The tree does not need to be rebalanced

Complexity: Logarithmic.

Throws: Nothing.

Note: This function will break container ordering invariants if new_node is not equivalent to node_to_be_replaced according to the ordering rules. This function is faster than erasing and inserting the node, since no rebalancing and comparison is needed. Experimental function

static void replace_node(node_ptr node_to_be_replaced, node_ptr header, 
                         node_ptr new_node) noexcept;

Requires: node_to_be_replaced must be inserted in a tree with header "header" and new_node must not be inserted in a tree.

Effects: Replaces node_to_be_replaced in its position in the tree with new_node. The tree does not need to be rebalanced

Complexity: Constant.

Throws: Nothing.

Note: This function will break container ordering invariants if new_node is not equivalent to node_to_be_replaced according to the ordering rules. This function is faster than erasing and inserting the node, since no rebalancing or comparison is needed. Experimental function

static node_ptr next_node(node_ptr n) noexcept;

Requires: 'n' is a node from the tree except the header.

Effects: Returns the next node of the tree.

Complexity: Average constant time.

Throws: Nothing.

static node_ptr prev_node(node_ptr n) noexcept;

Requires: 'n' is a node from the tree except the leftmost node.

Effects: Returns the previous node of the tree.

Complexity: Average constant time.

Throws: Nothing.

static node_ptr minimum(node_ptr n);

Requires: 'n' is a node of a tree but not the header.

Effects: Returns the minimum node of the subtree starting at p.

Complexity: Logarithmic to the size of the subtree.

Throws: Nothing.

static node_ptr maximum(node_ptr n);

Requires: 'n' is a node of a tree but not the header.

Effects: Returns the maximum node of the subtree starting at p.

Complexity: Logarithmic to the size of the subtree.

Throws: Nothing.

static void init(node_ptr n) noexcept;

Requires: 'n' must not be part of any tree.

Effects: After the function unique(node) == true.

Complexity: Constant.

Throws: Nothing.

Nodes: If node is inserted in a tree, this function corrupts the tree.

static bool inited(const_node_ptr n);

Effects: Returns true if node is in the same state as if called init(node)

Complexity: Constant.

Throws: Nothing.

static void init_header(node_ptr header) noexcept;

Requires: header must not be part of any tree.

Effects: Initializes the header to represent an empty tree. unique(header) == true.

Complexity: Constant.

Throws: Nothing.

Nodes: If header is inserted in a tree, this function corrupts the tree.

template<typename Disposer> 
  static void clear_and_dispose(node_ptr header, Disposer disposer) noexcept;

Requires: "disposer" must be an object function taking a node_ptr parameter and shouldn't throw.

Effects: Empties the target tree calling void disposer::operator()(node_ptr) for every node of the tree except the header.

Complexity: Linear to the number of element of the source tree plus the. number of elements of tree target tree when calling this function.

Throws: Nothing.

static node_ptr unlink_leftmost_without_rebalance(node_ptr header) noexcept;

Requires: header is the header of a tree.

Effects: Unlinks the leftmost node from the tree, and updates the header link to the new leftmost node.

Complexity: Average complexity is constant time.

Throws: Nothing.

Notes: This function breaks the tree and the tree can only be used for more unlink_leftmost_without_rebalance calls. This function is normally used to achieve a step by step controlled destruction of the tree.

static std::size_t size(const_node_ptr header) noexcept;

Requires: 'header' the header of the tree.

Effects: Returns the number of nodes of the tree.

Complexity: Linear time.

Throws: Nothing.

static void swap_tree(node_ptr header1, node_ptr header2) noexcept;

Requires: header1 and header2 must be the header nodes of two trees.

Effects: Swaps two trees. After the function header1 will contain links to the second tree and header2 will have links to the first tree.

Complexity: Constant.

Throws: Nothing.

static bool is_header(const_node_ptr p) noexcept;

Requires: p is a node of a tree.

Effects: Returns true if p is the header of the tree.

Complexity: Constant.

Throws: Nothing.

template<typename KeyType, typename KeyNodePtrCompare> 
  static node_ptr 
  find(const_node_ptr header, const KeyType & key, KeyNodePtrCompare comp);

Requires: "header" must be the header node of a tree. KeyNodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. KeyNodePtrCompare can compare KeyType with tree's node_ptrs.

Effects: Returns a node_ptr to the first element that is equivalent to "key" according to "comp" or "header" if that element does not exist.

Complexity: Logarithmic.

Throws: If "comp" throws.

template<typename KeyType, typename KeyNodePtrCompare> 
  static std::pair< node_ptr, node_ptr > 
  bounded_range(const_node_ptr header, const KeyType & lower_key, 
                const KeyType & upper_key, KeyNodePtrCompare comp, 
                bool left_closed, bool right_closed);

Requires: "header" must be the header node of a tree. KeyNodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. KeyNodePtrCompare can compare KeyType with tree's node_ptrs. 'lower_key' must not be greater than 'upper_key' according to 'comp'. If 'lower_key' == 'upper_key', ('left_closed' || 'right_closed') must be true.

Effects: Returns an a pair with the following criteria:

first = lower_bound(lower_key) if left_closed, upper_bound(lower_key) otherwise

second = upper_bound(upper_key) if right_closed, lower_bound(upper_key) otherwise

Complexity: Logarithmic.

Throws: If "comp" throws.

Note: This function can be more efficient than calling upper_bound and lower_bound for lower_key and upper_key.

Note: Experimental function, the interface might change.

template<typename KeyType, typename KeyNodePtrCompare> 
  static std::size_t 
  count(const_node_ptr header, const KeyType & key, KeyNodePtrCompare comp);

Requires: "header" must be the header node of a tree. KeyNodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. KeyNodePtrCompare can compare KeyType with tree's node_ptrs.

Effects: Returns the number of elements with a key equivalent to "key" according to "comp".

Complexity: Logarithmic.

Throws: If "comp" throws.

template<typename KeyType, typename KeyNodePtrCompare> 
  static std::pair< node_ptr, node_ptr > 
  equal_range(const_node_ptr header, const KeyType & key, 
              KeyNodePtrCompare comp);

Requires: "header" must be the header node of a tree. KeyNodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. KeyNodePtrCompare can compare KeyType with tree's node_ptrs.

Effects: Returns an a pair of node_ptr delimiting a range containing all elements that are equivalent to "key" according to "comp" or an empty range that indicates the position where those elements would be if there are no equivalent elements.

Complexity: Logarithmic.

Throws: If "comp" throws.

template<typename KeyType, typename KeyNodePtrCompare> 
  static std::pair< node_ptr, node_ptr > 
  lower_bound_range(const_node_ptr header, const KeyType & key, 
                    KeyNodePtrCompare comp);

Requires: "header" must be the header node of a tree. KeyNodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. KeyNodePtrCompare can compare KeyType with tree's node_ptrs.

Effects: Returns an a pair of node_ptr delimiting a range containing the first element that is equivalent to "key" according to "comp" or an empty range that indicates the position where that element would be if there are no equivalent elements.

Complexity: Logarithmic.

Throws: If "comp" throws.

template<typename KeyType, typename KeyNodePtrCompare> 
  static node_ptr 
  lower_bound(const_node_ptr header, const KeyType & key, 
              KeyNodePtrCompare comp);

Requires: "header" must be the header node of a tree. KeyNodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. KeyNodePtrCompare can compare KeyType with tree's node_ptrs.

Effects: Returns a node_ptr to the first element that is not less than "key" according to "comp" or "header" if that element does not exist.

Complexity: Logarithmic.

Throws: If "comp" throws.

template<typename KeyType, typename KeyNodePtrCompare> 
  static node_ptr 
  upper_bound(const_node_ptr header, const KeyType & key, 
              KeyNodePtrCompare comp);

Requires: "header" must be the header node of a tree. KeyNodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. KeyNodePtrCompare can compare KeyType with tree's node_ptrs.

Effects: Returns a node_ptr to the first element that is greater than "key" according to "comp" or "header" if that element does not exist.

Complexity: Logarithmic.

Throws: If "comp" throws.

static void insert_unique_commit(node_ptr header, node_ptr new_value, 
                                 const insert_commit_data & commit_data) noexcept;

Requires: "header" must be the header node of a tree. "commit_data" must have been obtained from a previous call to "insert_unique_check". No objects should have been inserted or erased from the set between the "insert_unique_check" that filled "commit_data" and the call to "insert_commit".

Effects: Inserts new_node in the set using the information obtained from the "commit_data" that a previous "insert_check" filled.

Complexity: Constant time.

Throws: Nothing.

Notes: This function has only sense if a "insert_unique_check" has been previously executed to fill "commit_data". No value should be inserted or erased between the "insert_check" and "insert_commit" calls.

template<typename KeyType, typename KeyNodePtrCompare> 
  static std::pair< node_ptr, bool > 
  insert_unique_check(const_node_ptr header, const KeyType & key, 
                      KeyNodePtrCompare comp, 
                      insert_commit_data & commit_data);

Requires: "header" must be the header node of a tree. KeyNodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. NodePtrCompare compares KeyType with a node_ptr.

Effects: Checks if there is an equivalent node to "key" in the tree according to "comp" and obtains the needed information to realize a constant-time node insertion if there is no equivalent node.

Returns: If there is an equivalent value returns a pair containing a node_ptr to the already present node and false. If there is not equivalent key can be inserted returns true in the returned pair's boolean and fills "commit_data" that is meant to be used with the "insert_commit" function to achieve a constant-time insertion function.

Complexity: Average complexity is at most logarithmic.

Throws: If "comp" throws.

Notes: This function is used to improve performance when constructing a node is expensive and the user does not want to have two equivalent nodes in the tree: if there is an equivalent value the constructed object must be discarded. Many times, the part of the node that is used to impose the order is much cheaper to construct than the node and this function offers the possibility to use that part to check if the insertion will be successful.

If the check is successful, the user can construct the node and use "insert_commit" to insert the node in constant-time. This gives a total logarithmic complexity to the insertion: check(O(log(N)) + commit(O(1)).

"commit_data" remains valid for a subsequent "insert_unique_commit" only if no more objects are inserted or erased from the set.

template<typename KeyType, typename KeyNodePtrCompare> 
  static std::pair< node_ptr, bool > 
  insert_unique_check(const_node_ptr header, node_ptr hint, 
                      const KeyType & key, KeyNodePtrCompare comp, 
                      insert_commit_data & commit_data);

Requires: "header" must be the header node of a tree. KeyNodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. NodePtrCompare compares KeyType with a node_ptr. "hint" is node from the "header"'s tree.

Effects: Checks if there is an equivalent node to "key" in the tree according to "comp" using "hint" as a hint to where it should be inserted and obtains the needed information to realize a constant-time node insertion if there is no equivalent node. If "hint" is the upper_bound the function has constant time complexity (two comparisons in the worst case).

Returns: If there is an equivalent value returns a pair containing a node_ptr to the already present node and false. If there is not equivalent key can be inserted returns true in the returned pair's boolean and fills "commit_data" that is meant to be used with the "insert_commit" function to achieve a constant-time insertion function.

Complexity: Average complexity is at most logarithmic, but it is amortized constant time if new_node should be inserted immediately before "hint".

Throws: If "comp" throws.

Notes: This function is used to improve performance when constructing a node is expensive and the user does not want to have two equivalent nodes in the tree: if there is an equivalent value the constructed object must be discarded. Many times, the part of the node that is used to impose the order is much cheaper to construct than the node and this function offers the possibility to use that part to check if the insertion will be successful.

If the check is successful, the user can construct the node and use "insert_commit" to insert the node in constant-time. This gives a total logarithmic complexity to the insertion: check(O(log(N)) + commit(O(1)).

"commit_data" remains valid for a subsequent "insert_unique_commit" only if no more objects are inserted or erased from the set.

template<typename NodePtrCompare> 
  static node_ptr 
  insert_equal(node_ptr h, node_ptr hint, node_ptr new_node, 
               NodePtrCompare comp);

Requires: "header" must be the header node of a tree. NodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. NodePtrCompare compares two node_ptrs. "hint" is node from the "header"'s tree.

Effects: Inserts new_node into the tree, using "hint" as a hint to where it will be inserted. If "hint" is the upper_bound the insertion takes constant time (two comparisons in the worst case).

Complexity: Logarithmic in general, but it is amortized constant time if new_node is inserted immediately before "hint".

Throws: If "comp" throws.

template<typename NodePtrCompare> 
  static node_ptr 
  insert_equal_upper_bound(node_ptr h, node_ptr new_node, NodePtrCompare comp);

Requires: "h" must be the header node of a tree. NodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. NodePtrCompare compares two node_ptrs.

Effects: Inserts new_node into the tree before the upper bound according to "comp".

Complexity: Average complexity for insert element is at most logarithmic.

Throws: If "comp" throws.

template<typename NodePtrCompare> 
  static node_ptr 
  insert_equal_lower_bound(node_ptr h, node_ptr new_node, NodePtrCompare comp);

Requires: "h" must be the header node of a tree. NodePtrCompare is a function object that induces a strict weak ordering compatible with the strict weak ordering used to create the the tree. NodePtrCompare compares two node_ptrs.

Effects: Inserts new_node into the tree before the lower bound according to "comp".

Complexity: Average complexity for insert element is at most logarithmic.

Throws: If "comp" throws.

static node_ptr 
insert_before(node_ptr header, node_ptr pos, node_ptr new_node) noexcept;

Requires: "header" must be the header node of a tree. "pos" must be a valid iterator or header (end) node. "pos" must be an iterator pointing to the successor to "new_node" once inserted according to the order of already inserted nodes. This function does not check "pos" and this precondition must be guaranteed by the caller.

Effects: Inserts new_node into the tree before "pos".

Complexity: Constant-time.

Throws: Nothing.

Note: If "pos" is not the successor of the newly inserted "new_node" tree invariants might be broken.

static void push_back(node_ptr header, node_ptr new_node) noexcept;

Requires: "header" must be the header node of a tree. "new_node" must be, according to the used ordering no less than the greatest inserted key.

Effects: Inserts new_node into the tree before "pos".

Complexity: Constant-time.

Throws: Nothing.

Note: If "new_node" is less than the greatest inserted key tree invariants are broken. This function is slightly faster than using "insert_before".

static void push_front(node_ptr header, node_ptr new_node) noexcept;

Requires: "header" must be the header node of a tree. "new_node" must be, according to the used ordering, no greater than the lowest inserted key.

Effects: Inserts new_node into the tree before "pos".

Complexity: Constant-time.

Throws: Nothing.

Note: If "new_node" is greater than the lowest inserted key tree invariants are broken. This function is slightly faster than using "insert_before".

static std::size_t depth(const_node_ptr n) noexcept;

Requires: 'n' can't be a header node.

Effects: Calculates the depth of a node: the depth of a node is the length (number of edges) of the path from the root to that node. (The root node is at depth 0.)

Complexity: Logarithmic to the number of nodes in the tree.

Throws: Nothing.

template<typename Cloner, typename Disposer> 
  static void clone(const_node_ptr source_header, node_ptr target_header, 
                    Cloner cloner, Disposer disposer);

Requires: "cloner" must be a function object taking a node_ptr and returning a new cloned node of it. "disposer" must take a node_ptr and shouldn't throw.

Effects: First empties target tree calling void disposer::operator()(node_ptr) for every node of the tree except the header.

Then, duplicates the entire tree pointed by "source_header" cloning each source node with node_ptr Cloner::operator()(node_ptr) to obtain the nodes of the target tree. If "cloner" throws, the cloned target nodes are disposed using void disposer(node_ptr ).

Complexity: Linear to the number of element of the source tree plus the number of elements of tree target tree when calling this function.

Throws: If cloner functor throws. If this happens target nodes are disposed.

static void erase(node_ptr header, node_ptr z) noexcept;

Requires: header must be the header of a tree, z a node of that tree and z != header.

Effects: Erases node "z" from the tree with header "header".

Complexity: Amortized constant time.

Throws: Nothing.

template<typename NodePtrCompare> 
  static bool transfer_unique(node_ptr header1, NodePtrCompare comp, 
                              node_ptr header2, node_ptr z);

Requires: header1 and header2 must be the headers of trees tree1 and tree2 respectively, z a non-header node of tree1. NodePtrCompare is the comparison function of tree1..

Effects: Transfers node "z" from tree1 to tree2 if tree1 does not contain a node that is equivalent to z.

Returns: True if the node was trasferred, false otherwise.

Complexity: Logarithmic.

Throws: If the comparison throws.

template<typename NodePtrCompare> 
  static void transfer_equal(node_ptr header1, NodePtrCompare comp, 
                             node_ptr header2, node_ptr z);

Requires: header1 and header2 must be the headers of trees tree1 and tree2 respectively, z a non-header node of tree1. NodePtrCompare is the comparison function of tree1..

Effects: Transfers node "z" from tree1 to tree2.

Complexity: Logarithmic.

Throws: If the comparison throws.

static void unlink(node_ptr n) noexcept;

Requires: 'n' is a tree node but not the header.

Effects: Unlinks the node and rebalances the tree.

Complexity: Average complexity is constant time.

Throws: Nothing.

static void rebalance(node_ptr header) noexcept;

Requires: header must be the header of a tree.

Effects: Rebalances the tree.

Throws: Nothing.

Complexity: Linear.

static node_ptr rebalance_subtree(node_ptr old_root) noexcept;

Requires: old_root is a node of a tree. It shall not be null.

Effects: Rebalances the subtree rooted at old_root.

Returns: The new root of the subtree.

Throws: Nothing.

Complexity: Linear.

template<typename Checker> 
  static void check(const_node_ptr header, Checker checker, 
                    typename Checker::return_type & checker_return);

Effects: Asserts the integrity of the container with additional checks provided by the user.

Requires: header must be the header of a tree.

Complexity: Linear time.

Note: The method might not have effect when asserts are turned off (e.g., with NDEBUG). Experimental function, interface might change in future versions.

template<typename NodePtrCompare> 
  static bool transfer_unique(node_ptr header1, NodePtrCompare comp, 
                              node_ptr header2, node_ptr z, 
                              data_for_rebalance & info);

template<typename NodePtrCompare> 
  static void transfer_equal(node_ptr header1, NodePtrCompare comp, 
                             node_ptr header2, node_ptr z, 
                             data_for_rebalance & info);

static void erase(node_ptr header, node_ptr z, data_for_rebalance & info);

static std::size_t subtree_size(const_node_ptr subtree) noexcept;

Requires: 'subtree' is a node of the tree but it's not the header.

Effects: Returns the number of nodes of the subtree.

Complexity: Linear time.

Throws: Nothing.

static bool is_left_child(node_ptr p) noexcept;

Requires: p is a node of a tree.

Effects: Returns true if p is a left child.

Complexity: Constant.

Throws: Nothing.

static bool is_right_child(node_ptr p) noexcept;

Requires: p is a node of a tree.

Effects: Returns true if p is a right child.

Complexity: Constant.

Throws: Nothing.

static void insert_before_check(node_ptr header, node_ptr pos, 
                                insert_commit_data & commit_data);

static void push_back_check(node_ptr header, insert_commit_data & commit_data) noexcept;

static void push_front_check(node_ptr header, 
                             insert_commit_data & commit_data) noexcept;

template<typename NodePtrCompare> 
  static void insert_equal_check(node_ptr header, node_ptr hint, 
                                 node_ptr new_node, NodePtrCompare comp, 
                                 insert_commit_data & commit_data);

template<typename NodePtrCompare> 
  static void insert_equal_upper_bound_check(node_ptr h, node_ptr new_node, 
                                             NodePtrCompare comp, 
                                             insert_commit_data & commit_data, 
                                             std::size_t * pdepth = 0);

template<typename NodePtrCompare> 
  static void insert_equal_lower_bound_check(node_ptr h, node_ptr new_node, 
                                             NodePtrCompare comp, 
                                             insert_commit_data & commit_data, 
                                             std::size_t * pdepth = 0);

static void insert_commit(node_ptr header, node_ptr new_node, 
                          const insert_commit_data & commit_data) noexcept;

static void set_child(node_ptr header, node_ptr new_child, 
                      node_ptr new_parent, const bool link_left) noexcept;

static void rotate_left_no_parent_fix(node_ptr p, node_ptr p_right) noexcept;

static void rotate_left(node_ptr p, node_ptr p_right, node_ptr p_parent, 
                        node_ptr header) noexcept;

static void rotate_right_no_parent_fix(node_ptr p, node_ptr p_left) noexcept;

static void rotate_right(node_ptr p, node_ptr p_left, node_ptr p_parent, 
                         node_ptr header) noexcept;

static void subtree_to_vine(node_ptr vine_tail, std::size_t & size) noexcept;

static void compress_subtree(node_ptr scanner, std::size_t count) noexcept;

static void vine_to_subtree(node_ptr super_root, std::size_t count) noexcept;

static node_ptr get_root(node_ptr n) noexcept;

Requires: "n" must be a node inserted in a tree.

Effects: Returns a pointer to the header node of the tree.

Complexity: Logarithmic.

Throws: Nothing.

template<typename Cloner, typename Disposer> 
  static node_ptr 
  clone_subtree(const_node_ptr source_parent, node_ptr target_parent, 
                Cloner cloner, Disposer disposer, node_ptr & leftmost_out, 
                node_ptr & rightmost_out);

template<typename Disposer> 
  static void dispose_subtree(node_ptr x, Disposer disposer) noexcept;

template<typename KeyType, typename KeyNodePtrCompare> 
  static node_ptr 
  lower_bound_loop(node_ptr x, node_ptr y, const KeyType & key, 
                   KeyNodePtrCompare comp);

template<typename KeyType, typename KeyNodePtrCompare> 
  static node_ptr 
  upper_bound_loop(node_ptr x, node_ptr y, const KeyType & key, 
                   KeyNodePtrCompare comp);

template<typename Checker> 
  static void check_subtree(const_node_ptr n, Checker checker, 
                            typename Checker::return_type & check_return);