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  <header>
    <copyright>
      <year>2001</year><year>2016</year>
      <holder>Ericsson AB. All Rights Reserved.</holder>
    </copyright>
    <legalnotice>
      Licensed under the Apache License, Version 2.0 (the "License");
      you may not use this file except in compliance with the License.
      You may obtain a copy of the License at
 
          http://www.apache.org/licenses/LICENSE-2.0

      Unless required by applicable law or agreed to in writing, software
      distributed under the License is distributed on an "AS IS" BASIS,
      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
      See the License for the specific language governing permissions and
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    </legalnotice>

    <title>gb_trees</title>
    <prepared></prepared>
    <docno></docno>
    <date></date>
    <rev></rev>
  </header>
  <module>gb_trees</module>
  <modulesummary>General balanced trees.</modulesummary>
  <description>
    <p>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.</p>

    <p>This module considers two keys as different if and only if
      they do not compare equal (<c>==</c>).</p>
  </description>

  <section>
    <title>Data Structure</title>
    <code type="none">
{Size, Tree}</code>

    <p><c>Tree</c> is composed of nodes of the form <c>{Key, Value, Smaller,
      Bigger}</c> and the "empty tree" node <c>nil</c>.</p>

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

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

  <datatypes>
    <datatype>
      <name name="tree" n_vars="2"/>
      <desc><p>A general balanced tree.</p></desc>
    </datatype>
    <datatype>
      <name name="tree" n_vars="0"/>
    </datatype>
    <datatype>
      <name name="iter" n_vars="2"/>
      <desc><p>A general balanced tree iterator.</p></desc>
    </datatype>
    <datatype>
      <name name="iter" n_vars="0"/>
    </datatype>
  </datatypes>

  <funcs>
    <func>
      <name name="balance" arity="1"/>
      <fsummary>Rebalance a tree.</fsummary>
      <desc>
        <p>Rebalances <c><anno>Tree1</anno></c>. 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.</p>
      </desc>
    </func>

    <func>
      <name name="delete" arity="2"/>
      <fsummary>Remove a node from a tree.</fsummary>
      <desc>
        <p>Removes the node with key <c><anno>Key</anno></c> from
          <c><anno>Tree1</anno></c> and returns the new tree. Assumes that the
          key is present in the tree, crashes otherwise.</p>
      </desc>
    </func>

    <func>
      <name name="delete_any" arity="2"/>
      <fsummary>Remove a (possibly non-existing) node from a tree.</fsummary>
      <desc>
        <p>Removes the node with key <c><anno>Key</anno></c> from
          <c><anno>Tree1</anno></c> if
          the key is present in the tree, otherwise does nothing.
          Returns the new tree.</p>
      </desc>
    </func>

    <func>
      <name name="empty" arity="0"/>
      <fsummary>Return an empty tree.</fsummary>
      <desc>
        <p>Returns a new empty tree.</p>
      </desc>
    </func>

    <func>
      <name name="enter" arity="3"/>
      <fsummary>Insert or update key with value in a tree.</fsummary>
      <desc>
        <p>Inserts <c><anno>Key</anno></c> with value <c><anno>Value</anno></c>
          into <c><anno>Tree1</anno></c> if the key is not present in the tree,
          otherwise updates <c><anno>Key</anno></c> to value
          <c><anno>Value</anno></c> in <c><anno>Tree1</anno></c>. Returns the
          new tree.</p>
      </desc>
    </func>

    <func>
      <name name="from_orddict" arity="1"/>
      <fsummary>Make a tree from an orddict.</fsummary>
      <desc>
        <p>Turns an ordered list <c><anno>List</anno></c> of key-value tuples
          into a tree. The list must not contain duplicate keys.</p>
      </desc>
    </func>

    <func>
      <name name="get" arity="2"/>
      <fsummary>Look up a key in a tree, if present.</fsummary>
      <desc>
        <p>Retrieves the value stored with <c><anno>Key</anno></c> in
          <c><anno>Tree</anno></c>.
          Assumes that the key is present in the tree, crashes
          otherwise.</p>
      </desc>
    </func>

    <func>
      <name name="insert" arity="3"/>
      <fsummary>Insert a new key and value in a tree.</fsummary>
      <desc>
        <p>Inserts <c><anno>Key</anno></c> with value <c><anno>Value</anno></c>
          into <c><anno>Tree1</anno></c> and
          returns the new tree. Assumes that the key is not present in
          the tree, crashes otherwise.</p>
      </desc>
    </func>

    <func>
      <name name="is_defined" arity="2"/>
      <fsummary>Test for membership of a tree.</fsummary>
      <desc>
        <p>Returns <c>true</c> if <c><anno>Key</anno></c> is present in
          <c><anno>Tree</anno></c>, otherwise <c>false</c>.</p>
      </desc>
    </func>

    <func>
      <name name="is_empty" arity="1"/>
      <fsummary>Test for empty tree.</fsummary>
      <desc>
        <p>Returns <c>true</c> if <c><anno>Tree</anno></c> is an empty tree,
          othwewise  <c>false</c>.</p>
      </desc>
    </func>

    <func>
      <name name="iterator" arity="1"/>
      <fsummary>Return an iterator for a tree.</fsummary>
      <desc>
        <p>Returns an iterator that can be used for traversing the
          entries of <c><anno>Tree</anno></c>; see
          <seealso marker="#next/1"><c>next/1</c></seealso>. The implementation
          of this is very efficient; traversing the whole tree using
          <c>next/1</c> is only slightly slower than getting the list
          of all elements using
          <seealso marker="#to_list/1"><c>to_list/1</c></seealso>
          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.</p>
      </desc>
    </func>

    <func>
      <name name="iterator_from" arity="2"/>
      <fsummary>Return an iterator for a tree starting from a specified key.
      </fsummary>
      <desc>
        <p>Returns an iterator that can be used for traversing the
          entries of <c><anno>Tree</anno></c>; see
          <seealso marker="#next/1"><c>next/1</c></seealso>.
          The difference as compared to the iterator returned by
          <seealso marker="#iterator/1"><c>iterator/1</c></seealso>
          is that the first key greater than
          or equal to <c><anno>Key</anno></c> is returned.</p>
      </desc>
    </func>

    <func>
      <name name="keys" arity="1"/>
      <fsummary>Return a list of the keys in a tree.</fsummary>
      <desc>
        <p>Returns the keys in <c><anno>Tree</anno></c> as an ordered list.</p>
      </desc>
    </func>

    <func>
      <name name="largest" arity="1"/>
      <fsummary>Return largest key and value.</fsummary>
      <desc>
        <p>Returns <c>{<anno>Key</anno>, <anno>Value</anno>}</c>, where
          <c><anno>Key</anno></c> is the largest
          key in <c><anno>Tree</anno></c>, and <c><anno>Value</anno></c> is
          the value associated
          with this key. Assumes that the tree is not empty.</p>
      </desc>
    </func>

    <func>
      <name name="lookup" arity="2"/>
      <fsummary>Look up a key in a tree.</fsummary>
      <desc>
        <p>Looks up <c><anno>Key</anno></c> in <c><anno>Tree</anno></c>.
          Returns <c>{value, <anno>Value</anno>}</c>, or <c>none</c> if
          <c><anno>Key</anno></c> is not present.</p>
      </desc>
    </func>

    <func>
      <name name="map" arity="2"/>
      <fsummary>Return largest key and value.</fsummary>
      <desc>
        <p>Maps function F(<anno>K</anno>, <anno>V1</anno>) -> <anno>V2</anno>
          to all key-value pairs of tree <c><anno>Tree1</anno></c>. Returns a
          new tree <c><anno>Tree2</anno></c> with the same set of keys as
	  <c><anno>Tree1</anno></c> and the new set of values
          <c><anno>V2</anno></c>.</p>
         </desc>
    </func>

    <func>
      <name name="next" arity="1"/>
      <fsummary>Traverse a tree with an iterator.</fsummary>
      <desc>
        <p>Returns <c>{<anno>Key</anno>, <anno>Value</anno>,
          <anno>Iter2</anno>}</c>, where <c><anno>Key</anno></c> is the
          smallest key referred to by iterator <c><anno>Iter1</anno></c>, and
          <c><anno>Iter2</anno></c> is the new iterator to be used for
          traversing the remaining nodes, or the atom <c>none</c> if no
          nodes remain.</p>
      </desc>
    </func>

    <func>
      <name name="size" arity="1"/>
      <fsummary>Return the number of nodes in a tree.</fsummary>
      <desc>
        <p>Returns the number of nodes in <c><anno>Tree</anno></c>.</p>
      </desc>
    </func>

    <func>
      <name name="smallest" arity="1"/>
      <fsummary>Return smallest key and value.</fsummary>
      <desc>
        <p>Returns <c>{<anno>Key</anno>, <anno>Value</anno>}</c>, where
          <c><anno>Key</anno></c> is the smallest
          key in <c><anno>Tree</anno></c>, and <c><anno>Value</anno></c> is
          the value associated
          with this key. Assumes that the tree is not empty.</p>
      </desc>
    </func>

    <func>
      <name name="take_largest" arity="1"/>
      <fsummary>Extract largest key and value.</fsummary>
      <desc>
        <p>Returns <c>{<anno>Key</anno>, <anno>Value</anno>,
          <anno>Tree2</anno>}</c>, where <c><anno>Key</anno></c> is the
          largest key in <c><anno>Tree1</anno></c>, <c><anno>Value</anno></c>
          is the value associated with this key, and <c><anno>Tree2</anno></c>
          is this tree with the corresponding node deleted. Assumes that the
          tree is not empty.</p>
      </desc>
    </func>

    <func>
      <name name="take_smallest" arity="1"/>
      <fsummary>Extract smallest key and value.</fsummary>
      <desc>
        <p>Returns <c>{<anno>Key</anno>, <anno>Value</anno>,
          <anno>Tree2</anno>}</c>, where <c><anno>Key</anno></c> is the
          smallest key in <c><anno>Tree1</anno></c>, <c><anno>Value</anno></c>
          is the value associated with this key, and <c><anno>Tree2</anno></c>
          is this tree with the corresponding node deleted. Assumes that the
          tree is not empty.</p>
      </desc>
    </func>

    <func>
      <name name="to_list" arity="1"/>
      <fsummary>Convert a tree into a list.</fsummary>
      <desc>
        <p>Converts a tree into an ordered list of key-value tuples.</p>
      </desc>
    </func>

    <func>
      <name name="update" arity="3"/>
      <fsummary>Update a key to new value in a tree.</fsummary>
      <desc>
        <p>Updates <c><anno>Key</anno></c> to value <c><anno>Value</anno></c>
          in <c><anno>Tree1</anno></c> and
          returns the new tree. Assumes that the key is present in the tree.</p>
      </desc>
    </func>

    <func>
      <name name="values" arity="1"/>
      <fsummary>Return a list of the values in a tree.</fsummary>
      <desc>
        <p>Returns the values in <c><anno>Tree</anno></c> as an ordered list,
          sorted by their corresponding keys. Duplicates are not removed.</p>
      </desc>
    </func>
  </funcs>

  <section>
    <title>See Also</title>
    <p><seealso marker="dict"><c>dict(3)</c></seealso>,
      <seealso marker="gb_sets"><c>gb_sets(3)</c></seealso></p>
  </section>
</erlref>

