-spec balance(Tree1) -> Tree2 when Tree1 :: tree(Key, Value), Tree2 :: tree(Key, Value).

Rebalances Tree1.

Notice that this is rarely necessary, but can be motivated when many nodes have been deleted from the tree without further insertions. Rebalancing can then be forced to minimize lookup times, as deletion does not rebalance the tree.

-spec delete(Key, Tree1) -> Tree2 when Tree1 :: tree(Key, Value), Tree2 :: tree(Key, Value).

Removes the node with key Key from Tree1 and returns the new tree. Assumes that the key is present in the tree, crashes otherwise.

-spec delete_any(Key, Tree1) -> Tree2 when Tree1 :: tree(Key, Value), Tree2 :: tree(Key, Value).

Removes the node with key Key from Tree1 if the key is present in the tree, otherwise does nothing. Returns the new tree.

-spec empty() -> tree(none(), none()).

Returns a new empty tree.

-spec enter(Key, Value, Tree1) -> Tree2 when Tree1 :: tree(Key, Value), Tree2 :: tree(Key, Value).

Inserts Key with value Value into Tree1 if the key is not present in the tree, otherwise updates Key to value Value in Tree1. Returns the new tree.

-spec from_orddict(List) -> Tree when List :: [{Key, Value}], Tree :: tree(Key, Value).

Turns an ordered list List of key-value tuples into a tree. The list must not contain duplicate keys.

-spec get(Key, Tree) -> Value when Tree :: tree(Key, Value).

Retrieves the value stored with Key in Tree. Assumes that the key is present in the tree, crashes otherwise.

-spec insert(Key, Value, Tree1) -> Tree2 when Tree1 :: tree(Key, Value), Tree2 :: tree(Key, Value).

Inserts Key with value Value into Tree1 and returns the new tree. Assumes that the key is not present in the tree, crashes otherwise.

-spec is_defined(Key, Tree) -> boolean() when Tree :: tree(Key, Value :: term()).

Returns true if Key is present in Tree, otherwise false.

-spec is_empty(Tree) -> boolean() when Tree :: tree().

Returns true if Tree is an empty tree, othwewise false.

-spec iterator(Tree) -> Iter when Tree :: tree(Key, Value), Iter :: iter(Key, Value).

Returns an iterator that can be used for traversing the entries of Tree; see next/1.

Equivalent to iterator(Tree, ordered).

-spec iterator(Tree, Order) -> Iter
                  when Tree :: tree(Key, Value), Iter :: iter(Key, Value), Order :: ordered | reversed.

Returns an iterator that can be used for traversing the entries of Tree in either ordered or reversed direction; see next/1.

The implementation of this is very efficient; traversing the whole tree using next/1 is only slightly slower than getting the list of all elements using to_list/1 and traversing that. The main advantage of the iterator approach is that it does not require the complete list of all elements to be built in memory at one time.

-spec iterator_from(Key, Tree) -> Iter when Tree :: tree(Key, Value), Iter :: iter(Key, Value).

Returns an iterator that can be used for traversing the entries of Tree; see next/1. The difference as compared to the iterator returned by iterator/1 is that the iterator starts with the first key greater than or equal to Key.

Equivalent to iterator_from(Key, Tree, ordered).

-spec iterator_from(Key, Tree, Order) -> Iter
                       when
                           Tree :: tree(Key, Value),
                           Iter :: iter(Key, Value),
                           Order :: ordered | reversed.

Returns an iterator that can be used for traversing the entries of Tree in either ordered or reversed direction; see next/1. The difference as compared to the iterator returned by iterator/2 is that the iterator starts with the first key next to or equal to Key.

-spec keys(Tree) -> [Key] when Tree :: tree(Key, Value :: term()).

Returns the keys in Tree as an ordered list.

-spec larger(Key1, Tree) -> none | {Key2, Value} when Key1 :: Key, Key2 :: Key, Tree :: tree(Key, Value).

Returns {Key2, Value}, where Key2 is the least key strictly greater than Key1, Value is the value associated with this key.

Returns none if no such pair exists.

-spec largest(Tree) -> {Key, Value} when Tree :: tree(Key, Value).

Returns {Key, Value}, where Key is the largest key in Tree, and Value is the value associated with this key. Assumes that the tree is not empty.

-spec lookup(Key, Tree) -> none | {value, Value} when Tree :: tree(Key, Value).

Looks up Key in Tree. Returns {value, Value}, or none if Key is not present.

-spec map(Function, Tree1) -> Tree2
             when
                 Function :: fun((K :: Key, V1 :: Value1) -> V2 :: Value2),
                 Tree1 :: tree(Key, Value1),
                 Tree2 :: tree(Key, Value2).

Maps function F(K, V1) -> V2 to all key-value pairs of tree Tree1. Returns a new tree Tree2 with the same set of keys as Tree1 and the new set of values V2.

-spec next(Iter1) -> none | {Key, Value, Iter2}
              when Iter1 :: iter(Key, Value), Iter2 :: iter(Key, Value).

Returns {Key, Value, Iter2}, where Key is the next key referred to by iterator Iter1, and Iter2 is the new iterator to be used for traversing the remaining nodes, or the atom none if no nodes remain.

-spec size(Tree) -> non_neg_integer() when Tree :: tree().

Returns the number of nodes in Tree.

-spec smaller(Key1, Tree) -> none | {Key2, Value}
                 when Key1 :: Key, Key2 :: Key, Tree :: tree(Key, Value).

Returns {Key2, Value}, where Key2 is the greatest key strictly less than Key1, Value is the value associated with this key.

Returns none if no such pair exists.

-spec smallest(Tree) -> {Key, Value} when Tree :: tree(Key, Value).

Returns {Key, Value}, where Key is the smallest key in Tree, and Value is the value associated with this key. Assumes that the tree is not empty.

-spec take(Key, Tree1) -> {Value, Tree2}
              when Tree1 :: tree(Key, _), Tree2 :: tree(Key, _), Key :: term(), Value :: term().

Returns a value Value from node with key Key and new Tree2 without the node with this value. Assumes that the node with key is present in the tree, crashes otherwise.

-spec take_any(Key, Tree1) -> {Value, Tree2} | error
                  when Tree1 :: tree(Key, _), Tree2 :: tree(Key, _), Key :: term(), Value :: term().

Returns a value Value from node with key Key and new Tree2 without the node with this value. Returns error if the node with the key is not present in the tree.

-spec take_largest(Tree1) -> {Key, Value, Tree2}
                      when Tree1 :: tree(Key, Value), Tree2 :: tree(Key, Value).

Returns {Key, Value, Tree2}, where Key is the largest key in Tree1, Value is the value associated with this key, and Tree2 is this tree with the corresponding node deleted. Assumes that the tree is not empty.

-spec take_smallest(Tree1) -> {Key, Value, Tree2}
                       when Tree1 :: tree(Key, Value), Tree2 :: tree(Key, Value).

Returns {Key, Value, Tree2}, where Key is the smallest key in Tree1, Value is the value associated with this key, and Tree2 is this tree with the corresponding node deleted. Assumes that the tree is not empty.

-spec to_list(Tree) -> [{Key, Value}] when Tree :: tree(Key, Value).

Converts a tree into an ordered list of key-value tuples.

-spec update(Key, Value, Tree1) -> Tree2 when Tree1 :: tree(Key, Value), Tree2 :: tree(Key, Value).

Updates Key to value Value in Tree1 and returns the new tree. Assumes that the key is present in the tree.

-spec values(Tree) -> [Value] when Tree :: tree(Key :: term(), Value).

Returns the values in Tree as an ordered list, sorted by their corresponding keys. Duplicates are not removed.