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TREE(3)		       FreeBSD Library Functions Manual		       TREE(3)

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_PROTOTYPE_INSERT, RB_PROTOTYPE_INSERT_COLOR,
     RB_PROTOTYPE_REMOVE, RB_PROTOTYPE_REMOVE_COLOR, RB_PROTOTYPE_FIND,
     RB_PROTOTYPE_NFIND, RB_PROTOTYPE_NEXT, RB_PROTOTYPE_PREV,
     RB_PROTOTYPE_MINMAX, RB_PROTOTYPE_REINSERT, RB_GENERATE,
     RB_GENERATE_STATIC, RB_GENERATE_INSERT, RB_GENERATE_INSERT_COLOR,
     RB_GENERATE_REMOVE, RB_GENERATE_REMOVE_COLOR, RB_GENERATE_FIND,
     RB_GENERATE_NFIND,	RB_GENERATE_NEXT, RB_GENERATE_PREV,
     RB_GENERATE_MINMAX, RB_GENERATE_REINSERT, 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_FROM, RB_FOREACH_SAFE, RB_FOREACH_REVERSE,
     RB_FOREACH_REVERSE_FROM, RB_FOREACH_REVERSE_SAFE, RB_INIT,	RB_INSERT,
     RB_REMOVE,	RB_REINSERT -- implementations of splay	and rank-balanced
     (wavl) trees

SYNOPSIS
     #include <sys/tree.h>

     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_PROTOTYPE_INSERT(NAME, TYPE, ATTR);

     RB_PROTOTYPE_INSERT_COLOR(NAME, TYPE, ATTR);

     RB_PROTOTYPE_REMOVE(NAME, TYPE, ATTR);

     RB_PROTOTYPE_REMOVE_COLOR(NAME, TYPE, ATTR);

     RB_PROTOTYPE_FIND(NAME, TYPE, ATTR);

     RB_PROTOTYPE_NFIND(NAME, TYPE, ATTR);

     RB_PROTOTYPE_NEXT(NAME, TYPE, ATTR);

     RB_PROTOTYPE_PREV(NAME, TYPE, ATTR);

     RB_PROTOTYPE_MINMAX(NAME, TYPE, ATTR);

     RB_PROTOTYPE_REINSERT(NAME, TYPE, ATTR);

     RB_GENERATE(NAME, TYPE, FIELD, CMP);

     RB_GENERATE_STATIC(NAME, TYPE, FIELD, CMP);

     RB_GENERATE_INSERT(NAME, TYPE, FIELD, CMP,	ATTR);

     RB_GENERATE_INSERT_COLOR(NAME, TYPE, FIELD, ATTR);

     RB_GENERATE_REMOVE(NAME, TYPE, FIELD, ATTR);

     RB_GENERATE_REMOVE_COLOR(NAME, TYPE, FIELD, ATTR);

     RB_GENERATE_FIND(NAME, TYPE, FIELD, CMP, ATTR);

     RB_GENERATE_NFIND(NAME, TYPE, FIELD, CMP, ATTR);

     RB_GENERATE_NEXT(NAME, TYPE, FIELD, ATTR);

     RB_GENERATE_PREV(NAME, TYPE, FIELD, ATTR);

     RB_GENERATE_MINMAX(NAME, TYPE, FIELD, ATTR);

     RB_GENERATE_REINSERT(NAME,	TYPE, FIELD, CMP, ATTR);

     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_FROM(VARNAME, NAME, POS_VARNAME);

     RB_FOREACH_SAFE(VARNAME, NAME, RB_HEAD *head, TEMP_VARNAME);

     RB_FOREACH_REVERSE(VARNAME, NAME, RB_HEAD *head);

     RB_FOREACH_REVERSE_FROM(VARNAME, NAME, POS_VARNAME);

     RB_FOREACH_REVERSE_SAFE(VARNAME, NAME, RB_HEAD *head, TEMP_VARNAME);

     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);

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

DESCRIPTION
     These macros define data structures for different types of	trees: splay
     trees and rank-balanced (wavl) trees.

     In	the macro definitions, TYPE is the name	tag of a user defined struc-
     ture 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	gener-
     ated 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 re-
     turns 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.

RANK-BALANCED TREES
     Rank-balanced (RB)	trees are a framework for defining height-balanced bi-
     nary search trees,	including AVL and red-black trees.  Each tree node has
     an	associated rank.  Balance conditions are expressed by conditions on
     the differences in	rank between any node and its children.	 Rank differ-
     ences are stored in each tree node.

     The balance conditions implemented	by the RB macros lead to weak AVL
     (wavl) trees, which combine the best aspects of AVL and red-black trees.
     Wavl trees	rebalance after	an insertion in	the same way AVL trees do,
     with the same worst-case time as red-black	trees offer, and with better
     balance in	the resulting tree.  Wavl trees	rebalance after	a removal in a
     way that requires less restructuring, in the worst	case, than either AVL
     or	red-black trees	do.  Removals can lead to a tree almost	as unbalanced
     as	a red-black tree; insertions lead to a tree becoming as	balanced as an
     AVL tree.

     A rank-balanced 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 con-
     nected 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().  Individual	prototypes can be declared with
     RB_PROTOTYPE_INSERT(), RB_PROTOTYPE_INSERT_COLOR(),
     RB_PROTOTYPE_REMOVE(), RB_PROTOTYPE_REMOVE_COLOR(), RB_PROTOTYPE_FIND(),
     RB_PROTOTYPE_NFIND(), RB_PROTOTYPE_NEXT(),	RB_PROTOTYPE_PREV(),
     RB_PROTOTYPE_MINMAX(), and	RB_PROTOTYPE_REINSERT()	in case	not all	func-
     tions are required.  The individual prototype macros expect NAME, TYPE,
     and ATTR arguments.  The ATTR argument must be empty for global functions
     or	static for static functions.

     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.  As an alternative individual function bodies are generated with
     the RB_GENERATE_INSERT(), RB_GENERATE_INSERT_COLOR(),
     RB_GENERATE_REMOVE(), RB_GENERATE_REMOVE_COLOR(), RB_GENERATE_FIND(),
     RB_GENERATE_NFIND(), RB_GENERATE_NEXT(), RB_GENERATE_PREV(),
     RB_GENERATE_MINMAX(), and RB_GENERATE_REINSERT() macros.

     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 re-
     turns 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 rank-balanced 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 ele-
     ment 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 macros	RB_FOREACH_SAFE() and RB_FOREACH_REVERSE_SAFE()	traverse the
     tree referenced by	head in	a forward or reverse direction respectively,
     assigning each element in turn to np.  However, unlike their unsafe coun-
     terparts, they permit both	the removal of np as well as freeing it	from
     within the	loop safely without interfering	with the traversal.

     Both RB_FOREACH_FROM() and	RB_FOREACH_REVERSE_FROM() may be used to con-
     tinue an interrupted traversal in a forward or reverse direction respec-
     tively.  The head pointer is not required.	 The pointer to	the node from
     where to resume the traversal should be passed as their last argument,
     and will be overwritten to	provide	safe traversal.

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

     The RB_REINSERT() macro updates the position of the element elm in	the
     tree.  This must be called	if a member of a tree is modified in a way
     that affects comparison, such as by modifying a node's key.  This is a
     lower overhead alternative	to removing the	element	and reinserting	it
     again.

EXAMPLES
     The following example demonstrates	how to declare a rank-balanced tree
     holding integers.	Values are inserted into it and	the contents of	the
     tree are printed in order.	 Lastly, the internal structure	of the tree is
     printed.

	#include <sys/tree.h>
	#include <err.h>
	#include <stdio.h>
	#include <stdlib.h>

	struct node {
		RB_ENTRY(node) entry;
		int i;
	};

	int
	intcmp(struct node *e1,	struct node *e2)
	{
		return (e1->i <	e2->i ?	-1 : e1->i > e2->i);
	}

	RB_HEAD(inttree, node) head = RB_INITIALIZER(&head);
	RB_GENERATE(inttree, node, entry, intcmp)

	int testdata[] = {
		20, 16,	17, 13,	3, 6, 1, 8, 2, 4, 10, 19, 5, 9,	12, 15,	18,
		7, 11, 14
	};

	void
	print_tree(struct node *n)
	{
		struct node *left, *right;

		if (n == NULL) {
			printf("nil");
			return;
		}
		left = RB_LEFT(n, entry);
		right =	RB_RIGHT(n, entry);
		if (left == NULL && right == NULL)
			printf("%d", n->i);
		else {
			printf("%d(", n->i);
			print_tree(left);
			printf(",");
			print_tree(right);
			printf(")");
		}
	}

	int
	main(void)
	{
		int i;
		struct node *n;

		for (i = 0; i <	sizeof(testdata) / sizeof(testdata[0]);	i++) {
			if ((n = malloc(sizeof(struct node))) == NULL)
				err(1, NULL);
			n->i = testdata[i];
			RB_INSERT(inttree, &head, n);
		}

		RB_FOREACH(n, inttree, &head) {
			printf("%d\n", n->i);
		}
		print_tree(RB_ROOT(&head));
		printf("\n");
		return (0);
	}

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	in-
     serted 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 re-
     moved element otherwise they return NULL to indicate an error.

SEE ALSO
     arb(3), queue(3)

     Bernhard Haeupler,	Siddhartha Sen,	and Robert E. Tarjan, "Rank-Balanced
     Trees", ACM Transactions on Algorithms, 4,	11,
     http://sidsen.azurewebsites.net/papers/rb-trees-talg.pdf, June 2015.

HISTORY
     The tree macros first appeared in FreeBSD 4.6.

AUTHORS
     The author	of the tree macros is Niels Provos.

FreeBSD	13.0			 July 27, 2020			  FreeBSD 13.0

NAME | SYNOPSIS | DESCRIPTION | SPLAY TREES | RANK-BALANCED TREES | EXAMPLES | NOTES | SEE ALSO | HISTORY | AUTHORS

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