## General balanced trees.

This module provides Prof. Arne Andersson'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 (`==`

).

#### Data Structure

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.

#### Types

### tree(Key, Value)

A general balanced tree.

### tree() = tree(term(), term())

### iter(Key, Value)

A general balanced tree iterator.

### iter() = iter(term(), term())

#### Functions

### balance(Tree1) -> Tree2

`Tree1 = Tree2 = tree(Key, Value)`

Rebalances

. 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

`Tree1 = Tree2 = tree(Key, Value)`

Removes the node with key

from

and returns the new tree. Assumes that the
key is present in the tree, crashes otherwise.

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

`Tree1 = Tree2 = tree(Key, Value)`

Removes the node with key

from

if
the key is present in the tree, otherwise does nothing.
Returns the new tree.

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

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

`Key = Value = term()`

Returns a value

from node with key

and new

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

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

`Key = Value = term()`

Returns a value

from node with key

and new

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

`Tree1 = Tree2 = tree(Key, Value)`

Inserts

with value

into

if the key is not present in the tree,
otherwise updates

to value

in

. Returns the
new tree.

### from_orddict(List) -> Tree

`List = [{Key, Value}]`

`Tree = tree(Key, Value)`

Turns an ordered list

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

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

`Tree = tree(Key, Value)`

Retrieves the value stored with

in

.
Assumes that the key is present in the tree, crashes
otherwise.

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

`Tree1 = Tree2 = tree(Key, Value)`

Inserts

with value

into

and
returns the new tree. Assumes that the key is not present in
the tree, crashes otherwise.

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

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

Returns `true`

if

is present in

, otherwise `false`

.

### iterator(Tree) -> Iter

Returns an iterator that can be used for traversing the
entries of

; 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

Returns an iterator that can be used for traversing the
entries of

; see
`next/1`

.
The difference as compared to the iterator returned by
`iterator/1`

is that the first key greater than
or equal to

is returned.

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

`Tree = tree(Key, Value)`

Returns `{`

, where

is the largest
key in

, and

is
the value associated
with this key. Assumes that the tree is not empty.

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

`Tree = tree(Key, Value)`

Looks up

in

.
Returns `{value, `

, or `none`

if

is not present.

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

`Function = fun((K :: Key, V1 :: Value1) -> V2 :: Value2)`

`Tree1 = tree(Key, Value1)`

`Tree2 = tree(Key, Value2)`

Maps function F(

. Returns a
new tree

with the same set of keys as

and the new set of values

.

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

`Iter1 = Iter2 = iter(Key, Value)`

Returns `{`

, where

is the
smallest key referred to by iterator

, and

is the new iterator to be used for
traversing the remaining nodes, or the atom `none`

if no
nodes remain.

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

`Tree = tree(Key, Value)`

Returns `{`

, where

is the smallest
key in

, and

is
the value associated
with this key. Assumes that the tree is not empty.

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

`Tree1 = Tree2 = tree(Key, Value)`

Returns `{`

, where

is the
largest key in

,

is the value associated with this key, and

is this tree with the corresponding node deleted. Assumes that the
tree is not empty.

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

`Tree1 = Tree2 = tree(Key, Value)`

Returns `{`

, where

is the
smallest key in

,

is the value associated with this key, and

is this tree with the corresponding node deleted. Assumes that the
tree is not empty.

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

`Tree = tree(Key, Value)`

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

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

`Tree1 = Tree2 = tree(Key, Value)`

Updates

to value

in

and
returns the new tree. Assumes that the key is present in the tree.

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

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

Returns the values in

as an ordered list,
sorted by their corresponding keys. Duplicates are not removed.