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TREE(3) Library Functions Manual TREE(3)

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

#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);

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 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 (), 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 (), 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.

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

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

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

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

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

The () 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 (), (), (), and () 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 () macro:

SPLAY_FOREACH(np, NAME, head)

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

Rank-balanced (RB) trees are a framework for defining height-balanced binary 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 differences 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 () 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 () 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 () or () 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 (), (), (), (), (), (), (), (), (), and () in case not all functions 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 () or () 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 (), (), (), (), (), (), (), (), (), and () 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 returns zero. Otherwise, it should return a value greater than zero. The compare function defines the order of the tree elements.

The () macro initializes the tree referenced by head.

The rank-balanced tree can also be initialized statically by using the () macro like this:

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

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

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

The () and () 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 (), (), (), (), and () 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 () or () macro:

RB_FOREACH(np, NAME, head)

The macros () and () traverse the tree referenced by head in a forward or reverse direction respectively, assigning each element in turn to np. However, unlike their unsafe counterparts, they permit both the removal of np as well as freeing it from within the loop safely without interfering with the traversal.

Both () and () may be used to continue an interrupted traversal in a forward or reverse direction respectively. 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 () macro should be used to check whether a rank-balanced tree is empty.

The () 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.

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);
}

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

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.

The tree macros first appeared in FreeBSD 4.6.

The author of the tree macros is Niels Provos.

July 27, 2020 FreeBSD-13.0