NAME

     SPLAY_PROTOTYPE, SPLAY_GENERATE, SPLAY_ENTRY, SPLAY_HEAD, SPLAY_INITIALIZER, SPLAY_ROOT,
     SPLAY_EMPTY, SPLAY_NEXT, SPLAY_MIN, SPLAY_MAX, SPLAY_FIND, SPLAY_LEFT, SPLAY_RIGHT,
     SPLAY_FOREACH, SPLAY_INIT, SPLAY_INSERT, SPLAY_REMOVE, RB_PROTOTYPE, RB_PROTOTYPE_STATIC,
     RB_GENERATE, RB_GENERATE_STATIC, RB_ENTRY, RB_HEAD, RB_INITIALIZER, RB_ROOT, RB_EMPTY,
     RB_NEXT, RB_PREV, RB_MIN, RB_MAX, RB_FIND, RB_NFIND, RB_LEFT, RB_RIGHT, RB_PARENT,
     RB_FOREACH, RB_FOREACH_REVERSE, RB_INIT, RB_INSERT, RB_REMOVE — implementations of splay and
     red-black trees

SYNOPSIS

     #include <sys/tree.h>
     (See libbsd(7) for include usage.)

     SPLAY_PROTOTYPE(NAME, TYPE, FIELD, CMP);

     SPLAY_GENERATE(NAME, TYPE, FIELD, CMP);

     SPLAY_ENTRY(TYPE);

     SPLAY_HEAD(HEADNAME, TYPE);

     struct TYPE *
     SPLAY_INITIALIZER(SPLAY_HEAD *head);

     SPLAY_ROOT(SPLAY_HEAD *head);

     bool
     SPLAY_EMPTY(SPLAY_HEAD *head);

     struct TYPE *
     SPLAY_NEXT(NAME, SPLAY_HEAD *head, struct TYPE *elm);

     struct TYPE *
     SPLAY_MIN(NAME, SPLAY_HEAD *head);

     struct TYPE *
     SPLAY_MAX(NAME, SPLAY_HEAD *head);

     struct TYPE *
     SPLAY_FIND(NAME, SPLAY_HEAD *head, struct TYPE *elm);

     struct TYPE *
     SPLAY_LEFT(struct TYPE *elm, SPLAY_ENTRY NAME);

     struct TYPE *
     SPLAY_RIGHT(struct TYPE *elm, SPLAY_ENTRY NAME);

     SPLAY_FOREACH(VARNAME, NAME, SPLAY_HEAD *head);

     void
     SPLAY_INIT(SPLAY_HEAD *head);

     struct TYPE *
     SPLAY_INSERT(NAME, SPLAY_HEAD *head, struct TYPE *elm);

     struct TYPE *
     SPLAY_REMOVE(NAME, SPLAY_HEAD *head, struct TYPE *elm);

     RB_PROTOTYPE(NAME, TYPE, FIELD, CMP);

     RB_PROTOTYPE_STATIC(NAME, TYPE, FIELD, CMP);

     RB_GENERATE(NAME, TYPE, FIELD, CMP);

     RB_GENERATE_STATIC(NAME, TYPE, FIELD, CMP);

     RB_ENTRY(TYPE);

     RB_HEAD(HEADNAME, TYPE);

     RB_INITIALIZER(RB_HEAD *head);

     struct TYPE *
     RB_ROOT(RB_HEAD *head);

     bool
     RB_EMPTY(RB_HEAD *head);

     struct TYPE *
     RB_NEXT(NAME, RB_HEAD *head, struct TYPE *elm);

     struct TYPE *
     RB_PREV(NAME, RB_HEAD *head, struct TYPE *elm);

     struct TYPE *
     RB_MIN(NAME, RB_HEAD *head);

     struct TYPE *
     RB_MAX(NAME, RB_HEAD *head);

     struct TYPE *
     RB_FIND(NAME, RB_HEAD *head, struct TYPE *elm);

     struct TYPE *
     RB_NFIND(NAME, RB_HEAD *head, struct TYPE *elm);

     struct TYPE *
     RB_LEFT(struct TYPE *elm, RB_ENTRY NAME);

     struct TYPE *
     RB_RIGHT(struct TYPE *elm, RB_ENTRY NAME);

     struct TYPE *
     RB_PARENT(struct TYPE *elm, RB_ENTRY NAME);

     RB_FOREACH(VARNAME, NAME, RB_HEAD *head);

     RB_FOREACH_REVERSE(VARNAME, NAME, RB_HEAD *head);

     void
     RB_INIT(RB_HEAD *head);

     struct TYPE *
     RB_INSERT(NAME, RB_HEAD *head, struct TYPE *elm);

     struct TYPE *
     RB_REMOVE(NAME, RB_HEAD *head, struct TYPE *elm);

DESCRIPTION

     These macros define data structures for different types of trees: splay trees and red-black
     trees.

     In the macro definitions, TYPE is the name tag of a user defined structure that must contain
     a field of type SPLAY_ENTRY, or RB_ENTRY, named ENTRYNAME.  The argument HEADNAME is the
     name tag of a user defined structure that must be declared using the macros SPLAY_HEAD(), or
     RB_HEAD().  The argument NAME has to be a unique name prefix for every tree that is defined.

     The function prototypes are declared with SPLAY_PROTOTYPE(), RB_PROTOTYPE(), or
     RB_PROTOTYPE_STATIC().  The function bodies are generated with SPLAY_GENERATE(),
     RB_GENERATE(), or RB_GENERATE_STATIC().  See the examples below for further explanation of
     how these macros are used.

SPLAY TREES

     A splay tree is a self-organizing data structure.  Every operation on the tree causes a
     splay to happen.  The splay moves the requested node to the root of the tree and partly
     rebalances it.

     This has the benefit that request locality causes faster lookups as the requested nodes move
     to the top of the tree.  On the other hand, every lookup causes memory writes.

     The Balance Theorem bounds the total access time for m operations and n inserts on an
     initially empty tree as O((m + n)lg n).  The amortized cost for a sequence of m accesses to
     a splay tree is O(lg n).

     A splay tree is headed by a structure defined by the SPLAY_HEAD() macro.  A structure is
     declared as follows:

           SPLAY_HEAD(HEADNAME, TYPE) head;

     where HEADNAME is the name of the structure to be defined, and struct TYPE is the type of
     the elements to be inserted into the tree.

     The SPLAY_ENTRY() macro declares a structure that allows elements to be connected in the
     tree.

     In order to use the functions that manipulate the tree structure, their prototypes need to
     be declared with the SPLAY_PROTOTYPE() macro, where NAME is a unique identifier for this
     particular tree.  The TYPE argument is the type of the structure that is being managed by
     the tree.  The FIELD argument is the name of the element defined by SPLAY_ENTRY().

     The function bodies are generated with the SPLAY_GENERATE() macro.  It takes the same
     arguments as the SPLAY_PROTOTYPE() macro, but should be used only once.

     Finally, the CMP argument is the name of a function used to compare tree nodes with each
     other.  The function takes two arguments of type struct TYPE *.  If the first argument is
     smaller than the second, the function returns a value smaller than zero.  If they are equal,
     the function returns zero.  Otherwise, it should return a value greater than zero.  The
     compare function defines the order of the tree elements.

     The SPLAY_INIT() macro initializes the tree referenced by head.

     The splay tree can also be initialized statically by using the SPLAY_INITIALIZER() macro
     like this:

           SPLAY_HEAD(HEADNAME, TYPE) head = SPLAY_INITIALIZER(&head);

     The SPLAY_INSERT() macro inserts the new element elm into the tree.

     The SPLAY_REMOVE() macro removes the element elm from the tree pointed by head.

     The SPLAY_FIND() macro can be used to find a particular element in the tree.

           struct TYPE find, *res;
           find.key = 30;
           res = SPLAY_FIND(NAME, head, &find);

     The SPLAY_ROOT(), SPLAY_MIN(), SPLAY_MAX(), and SPLAY_NEXT() macros can be used to traverse
     the tree:

           for (np = SPLAY_MIN(NAME, &head); np != NULL; np = SPLAY_NEXT(NAME, &head, np))

     Or, for simplicity, one can use the SPLAY_FOREACH() macro:

           SPLAY_FOREACH(np, NAME, head)

     The SPLAY_EMPTY() macro should be used to check whether a splay tree is empty.

RED-BLACK TREES

     A red-black tree is a binary search tree with the node color as an extra attribute.  It
     fulfills a set of conditions:

           1.   Every search path from the root to a leaf consists of the same number of black
                nodes.

           2.   Each red node (except for the root) has a black parent.

           3.   Each leaf node is black.

     Every operation on a red-black tree is bounded as O(lg n).  The maximum height of a red-
     black tree is 2lg(n + 1).

     A red-black tree is headed by a structure defined by the RB_HEAD() macro.  A structure is
     declared as follows:

           RB_HEAD(HEADNAME, TYPE) head;

     where HEADNAME is the name of the structure to be defined, and struct TYPE is the type of
     the elements to be inserted into the tree.

     The RB_ENTRY() macro declares a structure that allows elements to be connected in the tree.

     In order to use the functions that manipulate the tree structure, their prototypes need to
     be declared with the RB_PROTOTYPE() or RB_PROTOTYPE_STATIC() macro, where NAME is a unique
     identifier for this particular tree.  The TYPE argument is the type of the structure that is
     being managed by the tree.  The FIELD argument is the name of the element defined by
     RB_ENTRY().

     The function bodies are generated with the RB_GENERATE() or RB_GENERATE_STATIC() macro.
     These macros take the same arguments as the RB_PROTOTYPE() and RB_PROTOTYPE_STATIC() macros,
     but should be used only once.

     Finally, the CMP argument is the name of a function used to compare tree nodes with each
     other.  The function takes two arguments of type struct TYPE *.  If the first argument is
     smaller than the second, the function returns a value smaller than zero.  If they are equal,
     the function returns zero.  Otherwise, it should return a value greater than zero.  The
     compare function defines the order of the tree elements.

     The RB_INIT() macro initializes the tree referenced by head.

     The red-black tree can also be initialized statically by using the RB_INITIALIZER() macro
     like this:

           RB_HEAD(HEADNAME, TYPE) head = RB_INITIALIZER(&head);

     The RB_INSERT() macro inserts the new element elm into the tree.

     The RB_REMOVE() macro removes the element elm from the tree pointed by head.

     The RB_FIND() and RB_NFIND() macros can be used to find a particular element in the tree.

           struct TYPE find, *res;
           find.key = 30;
           res = RB_FIND(NAME, head, &find);

     The RB_ROOT(), RB_MIN(), RB_MAX(), RB_NEXT(), and RB_PREV() macros can be used to traverse
     the tree:

           for (np = RB_MIN(NAME, &head); np != NULL; np = RB_NEXT(NAME, &head, np))

     Or, for simplicity, one can use the RB_FOREACH() or RB_FOREACH_REVERSE() macro:

           RB_FOREACH(np, NAME, head)

     The RB_EMPTY() macro should be used to check whether a red-black tree is empty.

NOTES

     Trying to free a tree in the following way is a common error:

           SPLAY_FOREACH(var, NAME, head) {
                   SPLAY_REMOVE(NAME, head, var);
                   free(var);
           }
           free(head);

     Since var is freed, the FOREACH() macro refers to a pointer that may have been reallocated
     already.  Proper code needs a second variable.

           for (var = SPLAY_MIN(NAME, head); var != NULL; var = nxt) {
                   nxt = SPLAY_NEXT(NAME, head, var);
                   SPLAY_REMOVE(NAME, head, var);
                   free(var);
           }

     Both RB_INSERT() and SPLAY_INSERT() return NULL if the element was inserted in the tree
     successfully, otherwise they return a pointer to the element with the colliding key.

     Accordingly, RB_REMOVE() and SPLAY_REMOVE() return the pointer to the removed element
     otherwise they return NULL to indicate an error.

SEE ALSO

     queue(3bsd)

AUTHORS

     The author of the tree macros is Niels Provos.