PostgreSQL includes an implementation of the standard btree (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).
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 PostgreSQL'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 AM make use of them.
  As shown in Table 36.3, a btree operator
  class must provide five comparison operators,
  <,
  <=,
  =,
  >= and
  >.
  One might expect that <> should also be part of
  the operator class, but it is not, because it would almost never be
  useful to use a <> WHERE clause in an index
  search.  (For some purposes, the planner treats <>
  as associated with a btree operator class; but it finds that operator via
  the = operator's negator link, rather than
  from pg_amop.)
 
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 “loose” 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.
There are some basic assumptions that a btree operator family must satisfy:
    An = operator must be an equivalence relation; that
    is, for all non-null values A,
    B, C of the
    data type:
    
       A =
       A is true
       (reflexive law)
      
       if A =
       B,
       then B =
       A
       (symmetric law)
      
       if A =
       B and B
       = C,
       then A =
       C
       (transitive law)
      
    A < operator must be a strong ordering relation;
    that is, for all non-null values A,
    B, C:
    
       A <
       A is false
       (irreflexive law)
      
       if A <
       B
       and B <
       C,
       then A <
       C
       (transitive law)
      
    Furthermore, the ordering is total; that is, for all non-null
    values A, B:
    
       exactly one of A <
       B, A
       = B, and
       B <
       A is true
       (trichotomy law)
      
(The trichotomy law justifies the definition of the comparison support function, of course.)
  The other three operators are defined in terms of =
  and < in the obvious way, and must act consistently
  with them.
 
  For an operator family supporting multiple data types, the above laws must
  hold when A, B,
  C are taken from any data types in the family.
  The transitive laws are the trickiest to ensure, as in cross-type
  situations they represent statements that the behaviors of two or three
  different operators are consistent.
  As an example, it would not work to put float8
  and numeric into the same operator family, at least not with
  the current semantics that numeric values are converted
  to float8 for comparison to a float8.  Because
  of the limited accuracy of float8, this means there are
  distinct numeric values that will compare equal to the
  same float8 value, and thus the transitive law would fail.
 
Another requirement for a multiple-data-type family is that any implicit or binary-coercion casts that are defined between data types included in the operator family must not change the associated sort ordering.
It should be fairly clear why a btree index requires these laws to hold within a single data type: without them there is no ordering to arrange the keys with. Also, index searches using a comparison key of a different data type require comparisons to behave sanely across two data types. The extensions to three or more data types within a family are not strictly required by the btree index mechanism itself, but the planner relies on them for optimization purposes.
As shown in Table 36.9, btree defines one required and four optional support functions. The five user-defined methods are:
order
     For each combination of data types that a btree operator family
     provides comparison operators for, it must provide a comparison
     support function, registered in
     pg_amproc with support function number 1
     and
     amproclefttype/amprocrighttype
     equal to the left and right data types for the comparison (i.e.,
     the same data types that the matching operators are registered
     with in pg_amop).  The comparison
     function must take two non-null values
     A and B and
     return an int32 value that is
     < 0,
     0, or >
     0 when A
     < B,
     A =
     B, or A
     > B,
     respectively.  A null result is disallowed: all values of the
     data type must be comparable.  See
     src/backend/access/nbtree/nbtcompare.c for
     examples.
    
     If the compared values are of a collatable data type, the
     appropriate collation OID will be passed to the comparison
     support function, using the standard
     PG_GET_COLLATION() mechanism.
    
sortsupport
     Optionally, a btree operator family may provide sort
      support function(s), registered under support
     function number 2.  These functions allow implementing
     comparisons for sorting purposes in a more efficient way than
     naively calling the comparison support function.  The APIs
     involved in this are defined in
     src/include/utils/sortsupport.h.
    
in_range
     Optionally, a btree operator family may provide
     in_range support function(s), registered
     under support function number 3.  These are not used during btree
     index operations; rather, they extend the semantics of the
     operator family so that it can support window clauses containing
     the RANGE offset
     PRECEDING and RANGE
     offset FOLLOWING
     frame bound types (see Section 4.2.8).  Fundamentally, the extra
     information provided is how to add or subtract an
     offset value in a way that is
     compatible with the family's data ordering.
    
     An in_range function must have the signature
in_range(valtype1,basetype1,offsettype2,subbool,lessbool) returns bool
     val and
     base must be of the same type, which
     is one of the types supported by the operator family (i.e., a
     type for which it provides an ordering).  However,
     offset could be of a different type,
     which might be one otherwise unsupported by the family.  An
     example is that the built-in time_ops family
     provides an in_range function that has
     offset of type interval.
     A family can provide in_range functions for
     any of its supported types and one or more
     offset types.  Each
     in_range function should be entered in
     pg_amproc with
     amproclefttype equal to
     type1 and amprocrighttype
     equal to type2.
    
     The essential semantics of an in_range
     function depend on the two Boolean flag parameters.  It should
     add or subtract base and
     offset, then compare
     val to the result, as follows:
     
        if !sub and
        !less, return
        val >=
        (base +
        offset)
       
        if !sub and
        less, return
        val <=
        (base +
        offset)
       
        if sub and
        !less, return
        val >=
        (base -
        offset)
       
        if sub and
        less, return
        val <=
        (base -
        offset)
       
     Before doing so, the function should check the sign of
     offset: if it is less than zero, raise
     error
     ERRCODE_INVALID_PRECEDING_OR_FOLLOWING_SIZE
     (22013) with error text like “invalid preceding or
      following size in window function”.  (This is required by
     the SQL standard, although nonstandard operator families might
     perhaps choose to ignore this restriction, since there seems to
     be little semantic necessity for it.) This requirement is
     delegated to the in_range function so that
     the core code needn't understand what “less than
      zero” means for a particular data type.
    
     An additional expectation is that in_range
     functions should, if practical, avoid throwing an error if
     base +
     offset or
     base -
     offset would overflow.  The correct
     comparison result can be determined even if that value would be
     out of the data type's range.  Note that if the data type
     includes concepts such as “infinity” or
     “NaN”, extra care may be needed to ensure that
     in_range's results agree with the normal
     sort order of the operator family.
    
     The results of the in_range function must be
     consistent with the sort ordering imposed by the operator family.
     To be precise, given any fixed values of
     offset and
     sub, then:
     
        If in_range with
        less = true is true for some
        val1 and
        base, it must be true for every
        val2 <=
        val1 with the same
        base.
       
        If in_range with
        less = true is false for some
        val1 and
        base, it must be false for every
        val2 >=
        val1 with the same
        base.
       
        If in_range with
        less = true is true for some
        val and
        base1, it must be true for every
        base2 >=
        base1 with the same
        val.
       
        If in_range with
        less = true is false for some
        val and
        base1, it must be false for every
        base2 <=
        base1 with the same
        val.
       
     Analogous statements with inverted conditions hold when
     less = false.
    
     If the type being ordered (type1) is collatable, the
     appropriate collation OID will be passed to the
     in_range function, using the standard
     PG_GET_COLLATION() mechanism.
    
     in_range functions need not handle NULL
     inputs, and typically will be marked strict.
    
equalimage
     Optionally, a btree operator family may provide
     equalimage (“equality implies image
      equality”) support functions, registered under support
     function number 4.  These functions allow the core code to
     determine when it is safe to apply the btree deduplication
     optimization.  Currently, equalimage
     functions are only called when building or rebuilding an index.
    
     An equalimage function must have the
     signature
equalimage(opcintypeoid) returns bool
     The return value is static information about an operator class
     and collation.  Returning true indicates that
     the order function for the operator class is
     guaranteed to only return 0 (“arguments
      are equal”) when its A and
     B arguments are also interchangeable
     without any loss of semantic information.  Not registering an
     equalimage function or returning
     false indicates that this condition cannot be
     assumed to hold.
    
     The opcintype argument is the
     pg_type.oidequalimage function across operator classes.
     If opcintype is a collatable data
     type, the appropriate collation OID will be passed to the
     equalimage function, using the standard
     PG_GET_COLLATION() mechanism.
    
     As far as the operator class is concerned, returning
     true indicates that deduplication is safe (or
     safe for the collation whose OID was passed to its
     equalimage function).  However, the core
     code will only deem deduplication safe for an index when
     every indexed column uses an operator class
     that registers an equalimage function, and
     each function actually returns true when
     called.
    
     Image equality is almost the same condition
     as simple bitwise equality.  There is one subtle difference: When
     indexing a varlena data type, the on-disk representation of two
     image equal datums may not be bitwise equal due to inconsistent
     application of TOAST compression on input.
     Formally, when an operator class's
     equalimage function returns
     true, it is safe to assume that the
     datum_image_eq() C function will always agree
     with the operator class's order function
     (provided that the same collation OID is passed to both the
     equalimage and order
     functions).
    
     The core code is fundamentally unable to deduce anything about
     the “equality implies image equality” status of an
     operator class within a multiple-data-type family based on
     details from other operator classes in the same family.  Also, it
     is not sensible for an operator family to register a cross-type
     equalimage function, and attempting to do so
     will result in an error.  This is because “equality implies
      image equality” status does not just depend on
     sorting/equality semantics, which are more or less defined at the
     operator family level.  In general, the semantics that one
     particular data type implements must be considered separately.
    
     The convention followed by the operator classes included with the
     core PostgreSQL distribution is to
     register a stock, generic equalimage
     function.  Most operator classes register
     btequalimage(), which indicates that
     deduplication is safe unconditionally.  Operator classes for
     collatable data types such as text register
     btvarstrequalimage(), which indicates that
     deduplication is safe with deterministic collations.  Best
     practice for third-party extensions is to register their own
     custom function to retain control.
    
options
     Optionally, a B-tree operator family may provide
     options (“operator class specific
     options”) support functions, registered under support
     function number 5.  These functions define a set of user-visible
     parameters that control operator class behavior.
    
     An options support function must have the
     signature
options(reloptslocal_relopts *) returns void
     The function is passed a pointer to a local_relopts
     struct, which needs to be filled with a set of operator class
     specific options.  The options can be accessed from other support
     functions using the PG_HAS_OPCLASS_OPTIONS() and
     PG_GET_OPCLASS_OPTIONS() macros.
    
     Currently, no B-Tree operator class has an options
     support function.  B-tree doesn't allow flexible representation of keys
     like GiST, SP-GiST, GIN and BRIN do.  So, options
     probably doesn't have much application in the current B-tree index
     access method.  Nevertheless, this support function was added to B-tree
     for uniformity, and will probably find uses during further
     evolution of B-tree in PostgreSQL.
    
  This section covers B-Tree index implementation details that may be
  of use to advanced users.  See
  src/backend/access/nbtree/README in the source
  distribution for a much more detailed, internals-focused description
  of the B-Tree implementation.
 
PostgreSQL B-Tree indexes are multi-level tree structures, where each level of the tree can be used as a doubly-linked list of pages. A single metapage is stored in a fixed position at the start of the first segment file of the index. All other pages are either leaf pages or internal pages. Leaf pages are the pages on the lowest level of the tree. All other levels consist of internal pages. Each leaf page contains tuples that point to table rows. Each internal page contains tuples that point to the next level down in the tree. Typically, over 99% of all pages are leaf pages. Both internal pages and leaf pages use the standard page format described in Section 65.6.
New leaf pages are added to a B-Tree index when an existing leaf page cannot fit an incoming tuple. A page split operation makes room for items that originally belonged on the overflowing page by moving a portion of the items to a new page. Page splits must also insert a new downlink to the new page in the parent page, which may cause the parent to split in turn. Page splits “cascade upwards” in a recursive fashion. When the root page finally cannot fit a new downlink, a root page split operation takes place. This adds a new level to the tree structure by creating a new root page that is one level above the original root page.
   B-Tree indexes are not directly aware that under MVCC, there might
   be multiple extant versions of the same logical table row; to an
   index, each tuple is an independent object that needs its own index
   entry.  “Version churn” tuples may sometimes
   accumulate and adversely affect query latency and throughput.  This
   typically occurs with UPDATE-heavy workloads
   where most individual updates cannot apply the
   HOT optimization.
   Changing the value of only
   one column covered by one index during an UPDATE
   always necessitates a new set of index tuples
   — one for each and every index on the
   table.  Note in particular that this includes indexes that were not
   “logically modified” by the UPDATE.
   All indexes will need a successor physical index tuple that points
   to the latest version in the table.  Each new tuple within each
   index will generally need to coexist with the original
   “updated” tuple for a short period of time (typically
   until shortly after the UPDATE transaction
   commits).
  
   B-Tree indexes incrementally delete version churn index tuples by
   performing bottom-up index deletion passes.
   Each deletion pass is triggered in reaction to an anticipated
   “version churn page split”.  This only happens with
   indexes that are not logically modified by
   UPDATE statements, where concentrated build up
   of obsolete versions in particular pages would occur otherwise.  A
   page split will usually be avoided, though it's possible that
   certain implementation-level heuristics will fail to identify and
   delete even one garbage index tuple (in which case a page split or
   deduplication pass resolves the issue of an incoming new tuple not
   fitting on a leaf page).  The worst-case number of versions that
   any index scan must traverse (for any single logical row) is an
   important contributor to overall system responsiveness and
   throughput.  A bottom-up index deletion pass targets suspected
   garbage tuples in a single leaf page based on
   qualitative distinctions involving logical
   rows and versions.  This contrasts with the “top-down”
   index cleanup performed by autovacuum workers, which is triggered
   when certain quantitative table-level
   thresholds are exceeded (see Section 24.1.6).
  
    Not all deletion operations that are performed within B-Tree
    indexes are bottom-up deletion operations.  There is a distinct
    category of index tuple deletion: simple index tuple
     deletion.  This is a deferred maintenance operation
    that deletes index tuples that are known to be safe to delete
    (those whose item identifier's LP_DEAD bit is
    already set).  Like bottom-up index deletion, simple index
    deletion takes place at the point that a page split is anticipated
    as a way of avoiding the split.
   
    Simple deletion is opportunistic in the sense that it can only
    take place when recent index scans set the
    LP_DEAD bits of affected items in passing.
    Prior to PostgreSQL 14, the only
    category of B-Tree deletion was simple deletion.  The main
    differences between it and bottom-up deletion are that only the
    former is opportunistically driven by the activity of passing
    index scans, while only the latter specifically targets version
    churn from UPDATEs that do not logically modify
    indexed columns.
   
   Bottom-up index deletion performs the vast majority of all garbage
   index tuple cleanup for particular indexes with certain workloads.
   This is expected with any B-Tree index that is subject to
   significant version churn from UPDATEs that
   rarely or never logically modify the columns that the index covers.
   The average and worst-case number of versions per logical row can
   be kept low purely through targeted incremental deletion passes.
   It's quite possible that the on-disk size of certain indexes will
   never increase by even one single page/block despite
   constant version churn from
   UPDATEs.  Even then, an exhaustive “clean
    sweep” by a VACUUM operation (typically
   run in an autovacuum worker process) will eventually be required as
   a part of collective cleanup of the table and
   each of its indexes.
  
   Unlike VACUUM, bottom-up index deletion does not
   provide any strong guarantees about how old the oldest garbage
   index tuple may be.  No index can be permitted to retain
   “floating garbage” index tuples that became dead prior
   to a conservative cutoff point shared by the table and all of its
   indexes collectively.  This fundamental table-level invariant makes
   it safe to recycle table TIDs.  This is how it
   is possible for distinct logical rows to reuse the same table
   TID over time (though this can never happen with
   two logical rows whose lifetimes span the same
   VACUUM cycle).
  
A duplicate is a leaf page tuple (a tuple that points to a table row) where all indexed key columns have values that match corresponding column values from at least one other leaf page tuple in the same index. Duplicate tuples are quite common in practice. B-Tree indexes can use a special, space-efficient representation for duplicates when an optional technique is enabled: deduplication.
Deduplication works by periodically merging groups of duplicate tuples together, forming a single posting list tuple for each group. The column key value(s) only appear once in this representation. This is followed by a sorted array of TIDs that point to rows in the table. This significantly reduces the storage size of indexes where each value (or each distinct combination of column values) appears several times on average. The latency of queries can be reduced significantly. Overall query throughput may increase significantly. The overhead of routine index vacuuming may also be reduced significantly.
    B-Tree deduplication is just as effective with
    “duplicates” that contain a NULL value, even though
    NULL values are never equal to each other according to the
    = member of any B-Tree operator class.  As far
    as any part of the implementation that understands the on-disk
    B-Tree structure is concerned, NULL is just another value from the
    domain of indexed values.
   
The deduplication process occurs lazily, when a new item is inserted that cannot fit on an existing leaf page, though only when index tuple deletion could not free sufficient space for the new item (typically deletion is briefly considered and then skipped over). Unlike GIN posting list tuples, B-Tree posting list tuples do not need to expand every time a new duplicate is inserted; they are merely an alternative physical representation of the original logical contents of the leaf page. This design prioritizes consistent performance with mixed read-write workloads. Most client applications will at least see a moderate performance benefit from using deduplication. Deduplication is enabled by default.
   CREATE INDEX and REINDEX
   apply deduplication to create posting list tuples, though the
   strategy they use is slightly different.  Each group of duplicate
   ordinary tuples encountered in the sorted input taken from the
   table is merged into a posting list tuple
   before being added to the current pending leaf
   page.  Individual posting list tuples are packed with as many
   TIDs as possible.  Leaf pages are written out in
   the usual way, without any separate deduplication pass.  This
   strategy is well-suited to CREATE INDEX and
   REINDEX because they are once-off batch
   operations.
  
   Write-heavy workloads that don't benefit from deduplication due to
   having few or no duplicate values in indexes will incur a small,
   fixed performance penalty (unless deduplication is explicitly
   disabled).  The deduplicate_items storage
   parameter can be used to disable deduplication within individual
   indexes.  There is never any performance penalty with read-only
   workloads, since reading posting list tuples is at least as
   efficient as reading the standard tuple representation.  Disabling
   deduplication isn't usually helpful.
  
It is sometimes possible for unique indexes (as well as unique constraints) to use deduplication. This allows leaf pages to temporarily “absorb” extra version churn duplicates. Deduplication in unique indexes augments bottom-up index deletion, especially in cases where a long-running transaction holds a snapshot that blocks garbage collection. The goal is to buy time for the bottom-up index deletion strategy to become effective again. Delaying page splits until a single long-running transaction naturally goes away can allow a bottom-up deletion pass to succeed where an earlier deletion pass failed.
    A special heuristic is applied to determine whether a
    deduplication pass in a unique index should take place.  It can
    often skip straight to splitting a leaf page, avoiding a
    performance penalty from wasting cycles on unhelpful deduplication
    passes.  If you're concerned about the overhead of deduplication,
    consider setting deduplicate_items = off
    selectively.  Leaving deduplication enabled in unique indexes has
    little downside.
   
   Deduplication cannot be used in all cases due to
   implementation-level restrictions.  Deduplication safety is
   determined when CREATE INDEX or
   REINDEX is run.
  
Note that deduplication is deemed unsafe and cannot be used in the following cases involving semantically significant differences among equal datums:
      text, varchar, and char
      cannot use deduplication when a
      nondeterministic collation is used.  Case
      and accent differences must be preserved among equal datums.
     
      numeric cannot use deduplication.  Numeric display
      scale must be preserved among equal datums.
     
      jsonb cannot use deduplication, since the
      jsonb B-Tree operator class uses
      numeric internally.
     
      float4 and float8 cannot use
      deduplication.  These types have distinct representations for
      -0 and 0, which are
      nevertheless considered equal.  This difference must be
      preserved.
     
There is one further implementation-level restriction that may be lifted in a future version of PostgreSQL:
Container types (such as composite types, arrays, or range types) cannot use deduplication.
There is one further implementation-level restriction that applies regardless of the operator class or collation used:
      INCLUDE indexes can never use deduplication.