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<erlref>
  <header>
    <copyright>
      <year>2001</year><year>2011</year>
      <holder>Ericsson AB. All Rights Reserved.</holder>
    </copyright>
    <legalnotice>
      The contents of this file are subject to the Erlang Public License,
      Version 1.1, (the "License"); you may not use this file except in
      compliance with the License. You should have received a copy of the
      Erlang Public License along with this software. If not, it can be
      retrieved online at http://www.erlang.org/.

      Software distributed under the License is distributed on an "AS IS"
      basis, WITHOUT WARRANTY OF ANY KIND, either express or implied. See
      the License for the specific language governing rights and limitations
      under the License.

    </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>An efficient implementation of Prof. Arne Andersson's General
      Balanced Trees. These have no storage overhead compared to
      unbalanced binary trees, and their performance is in general
      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>
    <p>Data structure:</p>
    <code type="none">
      
- {Size, Tree}, where `Tree' is composed of nodes of the form:
  - {Key, Value, Smaller, Bigger}, and the "empty tree" node:
  - nil.</code>
    <p>There is no attempt to balance trees after deletions. Since
      deletions do not increase the height of a tree, this should be OK.</p>
    <p>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>
    <p>Performance is comparable to the AVL trees in the Erlang book
      (and faster in general due to less overhead); the difference is
      that deletion works for these trees, but not for the book's
      trees. Behaviour is logarithmic (as it should be).</p>
  </section>

  <datatypes>
    <datatype>
      <name><marker id="type-gb_tree">gb_tree()</marker></name>
      <desc><p>A GB tree.</p></desc>
    </datatype>
    <datatype>
      <name name="iter"/>
      <desc><p>A GB tree iterator.</p></desc>
    </datatype>
  </datatypes>
  <funcs>
    <func>
      <name name="balance" arity="1"/>
      <fsummary>Rebalance a tree</fsummary>
      <desc>
        <p>Rebalances <c><anno>Tree1</anno></c>. Note that this is rarely necessary,
          but may be motivated when a large number of nodes have been
          deleted from the tree without further insertions. Rebalancing
          could then be forced in order to minimise lookup times, since
          deletion only 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>;
          returns 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 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>Val</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>Val</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>Val</anno></c> into <c><anno>Tree1</anno></c>;
          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, and
          <c>false</c> otherwise.</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 <c>next/1</c>. 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 <c>to_list/1</c> 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="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>Val</anno>}</c>, where <c><anno>Key</anno></c> is the largest
          key in <c><anno>Tree</anno></c>, and <c><anno>Val</anno></c> is the value associated
          with this key. Assumes that the tree is nonempty.</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>Val</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 the function F(<anno>K</anno>, <anno>V1</anno>) -> <anno>V2</anno> to all key-value pairs
	    of the tree <c><anno>Tree1</anno></c> and 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>Val</anno>, <anno>Iter2</anno>}</c> where <c><anno>Key</anno></c> is the
          smallest key referred to by the 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>Val</anno>}</c>, where <c><anno>Key</anno></c> is the smallest
          key in <c><anno>Tree</anno></c>, and <c><anno>Val</anno></c> is the value associated
          with this key. Assumes that the tree is nonempty.</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>Val</anno>, <anno>Tree2</anno>}</c>, where <c><anno>Key</anno></c> is the
          largest key in <c><anno>Tree1</anno></c>, <c><anno>Val</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
          nonempty.</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>Val</anno>, <anno>Tree2</anno>}</c>, where <c><anno>Key</anno></c> is the
          smallest key in <c><anno>Tree1</anno></c>, <c><anno>Val</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
          nonempty.</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>Val</anno></c> in <c><anno>Tree1</anno></c>;
          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="gb_sets">gb_sets(3)</seealso>, 
      <seealso marker="dict">dict(3)</seealso></p>
  </section>
</erlref>

