gb_trees - General balanced trees.

This module provides Prof. Arne Anderssonâ€™s General Balanced Trees. These have no storage overhead compared to unbalanced binary trees, and their performance is better than AVL trees.

This module considers two keys as different if and only if they do not compare equal (*==*).

Trees and iterators are built using opaque data structures that should not be pattern-matched from outside this module.

There is no attempt to balance trees after deletions. As deletions do not increase the height of a tree, this should be OK.

The original balance condition *h(T) <= ceil(c * log(|T|))* has been changed to the similar (but not quite equivalent) condition *2 ^ h(T) <= |T| ^ c*. This should also be OK.

**tree(Key, Value)**

A general balanced tree.

**tree()** = tree(term(), term())

**iter(Key, Value)**

A general balanced tree iterator.

**iter()** = iter(term(), term())

**balance(Tree1) -> Tree2**

Types:

Tree1 = 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.

**delete(Key, Tree1) -> Tree2**

Types:

Tree1 = 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.

**delete_any(Key, Tree1) -> Tree2**

Types:

Tree1 = 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.

**take(Key, Tree1) -> {Value, Tree2}**

Types:

Tree1 = Tree2 = tree(Key, term())

Key = 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.

**take_any(Key, Tree1) -> {Value, Tree2} | error**

Types:

Tree1 = Tree2 = tree(Key, term())

Key = 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.

**empty() -> tree()**

Returns a new empty tree.

**enter(Key, Value, Tree1) -> Tree2**

Types:

Tree1 = 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.

**from_orddict(List) -> Tree**

Types:

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.

**get(Key, Tree) -> Value**

Types:

Tree = tree(Key, Value)

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

**insert(Key, Value, Tree1) -> Tree2**

Types:

Tree1 = 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.

**is_defined(Key, Tree) -> boolean()**

Types:

Tree = tree(Key, Value :: term())

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

**is_empty(Tree) -> boolean()**

Types:

Tree = tree()

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

**iterator(Tree) -> Iter**

Types:

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 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.

**iterator_from(Key, Tree) -> Iter**

Types:

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 first key greater than or equal to *Key* is returned.

**keys(Tree) -> [Key]**

Types:

Tree = tree(Key, Value :: term())

Returns the keys in *Tree* as an ordered list.

**largest(Tree) -> {Key, Value}**

Types:

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.

**lookup(Key, Tree) -> none | {value, Value}**

Types:

Tree = tree(Key, Value)

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

**map(Function, Tree1) -> Tree2**

Types:

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*.

**next(Iter1) -> none | {Key, Value, Iter2}**

Types:

Iter1 = Iter2 = iter(Key, Value)

Returns *{Key, Value, Iter2}*, where *Key* is the smallest 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.

**size(Tree) -> integer() >= 0**

Types:

Tree = tree()

Returns the number of nodes in *Tree*.

**smallest(Tree) -> {Key, Value}**

Types:

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.

**take_largest(Tree1) -> {Key, Value, Tree2}**

Types:

Tree1 = 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.

**take_smallest(Tree1) -> {Key, Value, Tree2}**

Types:

Tree1 = 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.

**to_list(Tree) -> [{Key, Value}]**

Types:

Tree = tree(Key, Value)

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

**update(Key, Value, Tree1) -> Tree2**

Types:

Tree1 = 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.

**values(Tree) -> [Value]**

Types:

Tree = tree(Key :: term(), Value)

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