<!-- doc/src/sgml/btree.sgml -->

<sect1 id="btree">
<title>B-Tree Indexes</title>

   <indexterm>
    <primary>index</primary>
    <secondary>B-Tree</secondary>
   </indexterm>

<sect2 id="btree-intro">
 <title>Introduction</title>

 <para>
  <productname>PostgreSQL</productname> includes an implementation of the
  standard <acronym>btree</acronym> (multi-way balanced tree) index data
  structure.  Any data type that can be sorted into a well-defined linear
  order can be indexed by a btree index.  The only limitation is that an
  index entry cannot exceed approximately one-third of a page (after TOAST
  compression, if applicable).
 </para>

 <para>
  Because each btree operator class imposes a sort order on its data type,
  btree operator classes (or, really, operator families) have come to be
  used as <productname>PostgreSQL</productname>'s general representation
  and understanding of sorting semantics.  Therefore, they've acquired
  some features that go beyond what would be needed just to support btree
  indexes, and parts of the system that are quite distant from the
  btree <acronym>AM</acronym> make use of them.
 </para>

</sect2>

<sect2 id="btree-behavior">
 <title>Behavior of B-Tree Operator Classes</title>

 <para>
  As shown in <xref linkend="xindex-btree-strat-table"/>, a btree operator
  class must provide five comparison operators,
  <literal>&lt;</literal>,
  <literal>&lt;=</literal>,
  <literal>=</literal>,
  <literal>&gt;=</literal> and
  <literal>&gt;</literal>.
  One might expect that <literal>&lt;&gt;</literal> should also be part of
  the operator class, but it is not, because it would almost never be
  useful to use a <literal>&lt;&gt;</literal> WHERE clause in an index
  search.  (For some purposes, the planner treats <literal>&lt;&gt;</literal>
  as associated with a btree operator class; but it finds that operator via
  the <literal>=</literal> operator's negator link, rather than
  from <structname>pg_amop</structname>.)
 </para>

 <para>
  When several data types share near-identical sorting semantics, their
  operator classes can be grouped into an operator family.  Doing so is
  advantageous because it allows the planner to make deductions about
  cross-type comparisons.  Each operator class within the family should
  contain the single-type operators (and associated support functions)
  for its input data type, while cross-type comparison operators and
  support functions are <quote>loose</quote> in the family.  It is
  recommendable that a complete set of cross-type operators be included
  in the family, thus ensuring that the planner can represent any
  comparison conditions that it deduces from transitivity.
 </para>

 <para>
  There are some basic assumptions that a btree operator family must
  satisfy:
 </para>

 <itemizedlist>
  <listitem>
   <para>
    An <literal>=</literal> operator must be an equivalence relation; that
    is, for all non-null values <replaceable>A</replaceable>,
    <replaceable>B</replaceable>, <replaceable>C</replaceable> of the
    data type:

    <itemizedlist>
     <listitem>
      <para>
       <replaceable>A</replaceable> <literal>=</literal>
       <replaceable>A</replaceable> is true
       (<firstterm>reflexive law</firstterm>)
      </para>
     </listitem>
     <listitem>
      <para>
       if <replaceable>A</replaceable> <literal>=</literal>
       <replaceable>B</replaceable>,
       then <replaceable>B</replaceable> <literal>=</literal>
       <replaceable>A</replaceable>
       (<firstterm>symmetric law</firstterm>)
      </para>
     </listitem>
     <listitem>
      <para>
       if <replaceable>A</replaceable> <literal>=</literal>
       <replaceable>B</replaceable> and <replaceable>B</replaceable>
       <literal>=</literal> <replaceable>C</replaceable>,
       then <replaceable>A</replaceable> <literal>=</literal>
       <replaceable>C</replaceable>
       (<firstterm>transitive law</firstterm>)
      </para>
     </listitem>
    </itemizedlist>