man/man3/tree.3

.\"	$OpenBSD: tree.3,v 1.7 2002/06/12 01:09:20 provos Exp $
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.\" Copyright 2002 Niels Provos <provos@citi.umich.edu>
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.Dd July 27, 2020
.Dt TREE 3
.Os
.Sh NAME
.Nm SPLAY_PROTOTYPE ,
.Nm SPLAY_GENERATE ,
.Nm SPLAY_ENTRY ,
.Nm SPLAY_HEAD ,
.Nm SPLAY_INITIALIZER ,
.Nm SPLAY_ROOT ,
.Nm SPLAY_EMPTY ,
.Nm SPLAY_NEXT ,
.Nm SPLAY_MIN ,
.Nm SPLAY_MAX ,
.Nm SPLAY_FIND ,
.Nm SPLAY_LEFT ,
.Nm SPLAY_RIGHT ,
.Nm SPLAY_FOREACH ,
.Nm SPLAY_INIT ,
.Nm SPLAY_INSERT ,
.Nm SPLAY_REMOVE ,
.Nm RB_PROTOTYPE ,
.Nm RB_PROTOTYPE_STATIC ,
.Nm RB_PROTOTYPE_INSERT ,
.Nm RB_PROTOTYPE_INSERT_COLOR ,
.Nm RB_PROTOTYPE_REMOVE ,
.Nm RB_PROTOTYPE_REMOVE_COLOR ,
.Nm RB_PROTOTYPE_FIND ,
.Nm RB_PROTOTYPE_NFIND ,
.Nm RB_PROTOTYPE_NEXT ,
.Nm RB_PROTOTYPE_PREV ,
.Nm RB_PROTOTYPE_MINMAX ,
.Nm RB_PROTOTYPE_REINSERT ,
.Nm RB_GENERATE ,
.Nm RB_GENERATE_STATIC ,
.Nm RB_GENERATE_INSERT ,
.Nm RB_GENERATE_INSERT_COLOR ,
.Nm RB_GENERATE_REMOVE ,
.Nm RB_GENERATE_REMOVE_COLOR ,
.Nm RB_GENERATE_FIND ,
.Nm RB_GENERATE_NFIND ,
.Nm RB_GENERATE_NEXT ,
.Nm RB_GENERATE_PREV ,
.Nm RB_GENERATE_MINMAX ,
.Nm RB_GENERATE_REINSERT ,
.Nm RB_ENTRY ,
.Nm RB_HEAD ,
.Nm RB_INITIALIZER ,
.Nm RB_ROOT ,
.Nm RB_EMPTY ,
.Nm RB_NEXT ,
.Nm RB_PREV ,
.Nm RB_MIN ,
.Nm RB_MAX ,
.Nm RB_FIND ,
.Nm RB_NFIND ,
.Nm RB_LEFT ,
.Nm RB_RIGHT ,
.Nm RB_PARENT ,
.Nm RB_FOREACH ,
.Nm RB_FOREACH_FROM ,
.Nm RB_FOREACH_SAFE ,
.Nm RB_FOREACH_REVERSE ,
.Nm RB_FOREACH_REVERSE_FROM ,
.Nm RB_FOREACH_REVERSE_SAFE ,
.Nm RB_INIT ,
.Nm RB_INSERT ,
.Nm RB_REMOVE ,
.Nm RB_REINSERT ,
.Nm RB_AUGMENT
.Nm RB_UPDATE_AUGMENT
.Nd "implementations of splay and rank-balanced (wavl) trees"
.Sh SYNOPSIS
.In sys/tree.h
.Fn SPLAY_PROTOTYPE NAME TYPE FIELD CMP
.Fn SPLAY_GENERATE NAME TYPE FIELD CMP
.Fn SPLAY_ENTRY TYPE
.Fn SPLAY_HEAD HEADNAME TYPE
.Ft "struct TYPE *"
.Fn SPLAY_INITIALIZER "SPLAY_HEAD *head"
.Fn SPLAY_ROOT "SPLAY_HEAD *head"
.Ft bool
.Fn SPLAY_EMPTY "SPLAY_HEAD *head"
.Ft "struct TYPE *"
.Fn SPLAY_NEXT NAME "SPLAY_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn SPLAY_MIN NAME "SPLAY_HEAD *head"
.Ft "struct TYPE *"
.Fn SPLAY_MAX NAME "SPLAY_HEAD *head"
.Ft "struct TYPE *"
.Fn SPLAY_FIND NAME "SPLAY_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn SPLAY_LEFT "struct TYPE *elm" "SPLAY_ENTRY NAME"
.Ft "struct TYPE *"
.Fn SPLAY_RIGHT "struct TYPE *elm" "SPLAY_ENTRY NAME"
.Fn SPLAY_FOREACH VARNAME NAME "SPLAY_HEAD *head"
.Ft void
.Fn SPLAY_INIT "SPLAY_HEAD *head"
.Ft "struct TYPE *"
.Fn SPLAY_INSERT NAME "SPLAY_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn SPLAY_REMOVE NAME "SPLAY_HEAD *head" "struct TYPE *elm"
.Fn RB_PROTOTYPE NAME TYPE FIELD CMP
.Fn RB_PROTOTYPE_STATIC NAME TYPE FIELD CMP
.Fn RB_PROTOTYPE_INSERT NAME TYPE ATTR
.Fn RB_PROTOTYPE_INSERT_COLOR NAME TYPE ATTR
.Fn RB_PROTOTYPE_REMOVE NAME TYPE ATTR
.Fn RB_PROTOTYPE_REMOVE_COLOR NAME TYPE ATTR
.Fn RB_PROTOTYPE_FIND NAME TYPE ATTR
.Fn RB_PROTOTYPE_NFIND NAME TYPE ATTR
.Fn RB_PROTOTYPE_NEXT NAME TYPE ATTR
.Fn RB_PROTOTYPE_PREV NAME TYPE ATTR
.Fn RB_PROTOTYPE_MINMAX NAME TYPE ATTR
.Fn RB_PROTOTYPE_REINSERT NAME TYPE ATTR
.Fn RB_GENERATE NAME TYPE FIELD CMP
.Fn RB_GENERATE_STATIC NAME TYPE FIELD CMP
.Fn RB_GENERATE_INSERT NAME TYPE FIELD CMP ATTR
.Fn RB_GENERATE_INSERT_COLOR NAME TYPE FIELD ATTR
.Fn RB_GENERATE_REMOVE NAME TYPE FIELD ATTR
.Fn RB_GENERATE_REMOVE_COLOR NAME TYPE FIELD ATTR
.Fn RB_GENERATE_FIND NAME TYPE FIELD CMP ATTR
.Fn RB_GENERATE_NFIND NAME TYPE FIELD CMP ATTR
.Fn RB_GENERATE_NEXT NAME TYPE FIELD ATTR
.Fn RB_GENERATE_PREV NAME TYPE FIELD ATTR
.Fn RB_GENERATE_MINMAX NAME TYPE FIELD ATTR
.Fn RB_GENERATE_REINSERT NAME TYPE FIELD CMP ATTR
.Fn RB_ENTRY TYPE
.Fn RB_HEAD HEADNAME TYPE
.Fn RB_INITIALIZER "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_ROOT "RB_HEAD *head"
.Ft "bool"
.Fn RB_EMPTY "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_NEXT NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_PREV NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_MIN NAME "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_MAX NAME "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_FIND NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_NFIND NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_LEFT "struct TYPE *elm" "RB_ENTRY NAME"
.Ft "struct TYPE *"
.Fn RB_RIGHT "struct TYPE *elm" "RB_ENTRY NAME"
.Ft "struct TYPE *"
.Fn RB_PARENT "struct TYPE *elm" "RB_ENTRY NAME"
.Fn RB_FOREACH VARNAME NAME "RB_HEAD *head"
.Fn RB_FOREACH_FROM "VARNAME" "NAME" "POS_VARNAME"
.Fn RB_FOREACH_SAFE "VARNAME" "NAME" "RB_HEAD *head" "TEMP_VARNAME"
.Fn RB_FOREACH_REVERSE VARNAME NAME "RB_HEAD *head"
.Fn RB_FOREACH_REVERSE_FROM "VARNAME" "NAME" "POS_VARNAME"
.Fn RB_FOREACH_REVERSE_SAFE "VARNAME" "NAME" "RB_HEAD *head" "TEMP_VARNAME"
.Ft void
.Fn RB_INIT "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_INSERT NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_REMOVE NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_REINSERT NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "void"
.Fn RB_AUGMENT NAME "struct TYPE *elm"
.Ft "void"
.Fn RB_UPDATE_AUGMENT NAME "struct TYPE *elm"
.Sh DESCRIPTION
These macros define data structures for different types of trees:
splay trees and rank-balanced (wavl) trees.
.Pp
In the macro definitions,
.Fa TYPE
is the name tag of a user defined structure that must contain a field of type
.Vt SPLAY_ENTRY ,
or
.Vt RB_ENTRY ,
named
.Fa ENTRYNAME .
The argument
.Fa HEADNAME
is the name tag of a user defined structure that must be declared
using the macros
.Fn SPLAY_HEAD ,
or
.Fn RB_HEAD .
The argument
.Fa NAME
has to be a unique name prefix for every tree that is defined.
.Pp
The function prototypes are declared with
.Fn SPLAY_PROTOTYPE ,
.Fn RB_PROTOTYPE ,
or
.Fn RB_PROTOTYPE_STATIC .
The function bodies are generated with
.Fn SPLAY_GENERATE ,
.Fn RB_GENERATE ,
or
.Fn RB_GENERATE_STATIC .
See the examples below for further explanation of how these macros are used.
.Sh 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.
.Pp
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.
.Pp
The Balance Theorem bounds the total access time for
.Ar m
operations and
.Ar n
inserts on an initially empty tree as
.Fn O "\*[lp]m + n\*[rp]lg n" .
The
amortized cost for a sequence of
.Ar m
accesses to a splay tree is
.Fn O "lg n" .
.Pp
A splay tree is headed by a structure defined by the
.Fn SPLAY_HEAD
macro.
A
structure is declared as follows:
.Bd -ragged -offset indent
.Fn SPLAY_HEAD HEADNAME TYPE
.Va head ;
.Ed
.Pp
where
.Fa HEADNAME
is the name of the structure to be defined, and struct
.Fa TYPE
is the type of the elements to be inserted into the tree.
.Pp
The
.Fn SPLAY_ENTRY
macro declares a structure that allows elements to be connected in the tree.
.Pp
In order to use the functions that manipulate the tree structure,
their prototypes need to be declared with the
.Fn SPLAY_PROTOTYPE
macro,
where
.Fa NAME
is a unique identifier for this particular tree.
The
.Fa TYPE
argument is the type of the structure that is being managed
by the tree.
The
.Fa FIELD
argument is the name of the element defined by
.Fn SPLAY_ENTRY .
.Pp
The function bodies are generated with the
.Fn SPLAY_GENERATE
macro.
It takes the same arguments as the
.Fn SPLAY_PROTOTYPE
macro, but should be used only once.
.Pp
Finally,
the
.Fa CMP
argument is the name of a function used to compare tree nodes
with each other.
The function takes two arguments of type
.Vt "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.
.Pp
The
.Fn SPLAY_INIT
macro initializes the tree referenced by
.Fa head .
.Pp
The splay tree can also be initialized statically by using the
.Fn SPLAY_INITIALIZER
macro like this:
.Bd -ragged -offset indent
.Fn SPLAY_HEAD HEADNAME TYPE
.Va head
=
.Fn SPLAY_INITIALIZER &head ;
.Ed
.Pp
The
.Fn SPLAY_INSERT
macro inserts the new element
.Fa elm
into the tree.
.Pp
The
.Fn SPLAY_REMOVE
macro removes the element
.Fa elm
from the tree pointed by
.Fa head .
.Pp
The
.Fn SPLAY_FIND
macro can be used to find a particular element in the tree.
.Bd -literal -offset indent
struct TYPE find, *res;
find.key = 30;
res = SPLAY_FIND(NAME, head, &find);
.Ed
.Pp
The
.Fn SPLAY_ROOT ,
.Fn SPLAY_MIN ,
.Fn SPLAY_MAX ,
and
.Fn SPLAY_NEXT
macros can be used to traverse the tree:
.Bd -literal -offset indent
for (np = SPLAY_MIN(NAME, &head); np != NULL; np = SPLAY_NEXT(NAME, &head, np))
.Ed
.Pp
Or, for simplicity, one can use the
.Fn SPLAY_FOREACH
macro:
.Bd -ragged -offset indent
.Fn SPLAY_FOREACH np NAME head
.Ed
.Pp
The
.Fn SPLAY_EMPTY
macro should be used to check whether a splay tree is empty.
.Sh RANK-BALANCED TREES
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.
.Pp
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.
.Pp
A rank-balanced tree is headed by a structure defined by the
.Fn RB_HEAD
macro.
A
structure is declared as follows:
.Bd -ragged -offset indent
.Fn RB_HEAD HEADNAME TYPE
.Va head ;
.Ed
.Pp
where
.Fa HEADNAME
is the name of the structure to be defined, and struct
.Fa TYPE
is the type of the elements to be inserted into the tree.
.Pp
The
.Fn RB_ENTRY
macro declares a structure that allows elements to be connected in the tree.
.Pp
In order to use the functions that manipulate the tree structure,
their prototypes need to be declared with the
.Fn RB_PROTOTYPE
or
.Fn RB_PROTOTYPE_STATIC
macro,
where
.Fa NAME
is a unique identifier for this particular tree.
The
.Fa TYPE
argument is the type of the structure that is being managed
by the tree.
The
.Fa FIELD
argument is the name of the element defined by
.Fn RB_ENTRY .
Individual prototypes can be declared with
.Fn RB_PROTOTYPE_INSERT ,
.Fn RB_PROTOTYPE_INSERT_COLOR ,
.Fn RB_PROTOTYPE_REMOVE ,
.Fn RB_PROTOTYPE_REMOVE_COLOR ,
.Fn RB_PROTOTYPE_FIND ,
.Fn RB_PROTOTYPE_NFIND ,
.Fn RB_PROTOTYPE_NEXT ,
.Fn RB_PROTOTYPE_PREV ,
.Fn RB_PROTOTYPE_MINMAX ,
and
.Fn RB_PROTOTYPE_REINSERT
in case not all functions are required.
The individual prototype macros expect
.Fa NAME ,
.Fa TYPE ,
and
.Fa ATTR
arguments.
The
.Fa ATTR
argument must be empty for global functions or
.Fa static
for static functions.
.Pp
The function bodies are generated with the
.Fn RB_GENERATE
or
.Fn RB_GENERATE_STATIC
macro.
These macros take the same arguments as the
.Fn RB_PROTOTYPE
and
.Fn RB_PROTOTYPE_STATIC
macros, but should be used only once.
As an alternative individual function bodies are generated with the
.Fn RB_GENERATE_INSERT ,
.Fn RB_GENERATE_INSERT_COLOR ,
.Fn RB_GENERATE_REMOVE ,
.Fn RB_GENERATE_REMOVE_COLOR ,
.Fn RB_GENERATE_FIND ,
.Fn RB_GENERATE_NFIND ,
.Fn RB_GENERATE_NEXT ,
.Fn RB_GENERATE_PREV ,
.Fn RB_GENERATE_MINMAX ,
and
.Fn RB_GENERATE_REINSERT
macros.
.Pp
Finally,
the
.Fa CMP
argument is the name of a function used to compare tree nodes
with each other.
The function takes two arguments of type
.Vt "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.
.Pp
The
.Fn RB_INIT
macro initializes the tree referenced by
.Fa head .
.Pp
The rank-balanced tree can also be initialized statically by using the
.Fn RB_INITIALIZER
macro like this:
.Bd -ragged -offset indent
.Fn RB_HEAD HEADNAME TYPE
.Va head
=
.Fn RB_INITIALIZER &head ;
.Ed
.Pp
The
.Fn RB_INSERT
macro inserts the new element
.Fa elm
into the tree.
.Pp
The
.Fn RB_REMOVE
macro removes the element
.Fa elm
from the tree pointed by
.Fa head .
.Pp
The
.Fn RB_FIND
and
.Fn RB_NFIND
macros can be used to find a particular element in the tree.
.Bd -literal -offset indent
struct TYPE find, *res;
find.key = 30;
res = RB_FIND(NAME, head, &find);
.Ed
.Pp
The
.Fn RB_ROOT ,
.Fn RB_MIN ,
.Fn RB_MAX ,
.Fn RB_NEXT ,
and
.Fn RB_PREV
macros can be used to traverse the tree:
.Pp
.Dl "for (np = RB_MIN(NAME, &head); np != NULL; np = RB_NEXT(NAME, &head, np))"
.Pp
Or, for simplicity, one can use the
.Fn RB_FOREACH
or
.Fn RB_FOREACH_REVERSE
macro:
.Bd -ragged -offset indent
.Fn RB_FOREACH np NAME head
.Ed
.Pp
The macros
.Fn RB_FOREACH_SAFE
and
.Fn 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 counterparts,
they permit both the removal of np
as well as freeing it from within the loop safely
without interfering with the traversal.
.Pp
Both
.Fn RB_FOREACH_FROM
and
.Fn RB_FOREACH_REVERSE_FROM
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.
.Pp
The
.Fn RB_EMPTY
macro should be used to check whether a rank-balanced tree is empty.
.Pp
The
.Fn RB_REINSERT
macro updates the position of the element
.Fa elm
in the tree.
This must be called if a member of a
.Nm 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.
.Pp
The
.Fn RB_AUGMENT
macro updates augmentation data of the element
.Fa elm
in the tree.
By default, it has no effect.
It is not meant to be invoked by the RB user.
If
.Fn RB_AUGMENT
is defined by the RB user, then when an element is inserted or removed
from the tree, it is invoked for every element in the tree that is the
root of an altered subtree, working from the bottom of the tree up to
the top.
It is typically used to maintain some associative accumulation of tree
elements, such as sums, minima, maxima, and the like.
.Pp
The
.Fn RB_UPDATE_AUGMENT
macro updates augmentation data of the element
.Fa elm
and its ancestors in the tree.
If RB_AUGMENT is defined by the RB user, then when an element in the
tree is changed, without changing the order of items in the tree,
invoking this function on that element restores consistency of the
augmentation state of the tree as if the element had been removed and
inserted again.
.Sh 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.
To maintain the sum of the values in the tree, each element maintains
the sum of its value and the sums from its left and right subtrees.
Lastly, the internal structure of the tree is printed.
.Bd -literal -offset 3n
#include <sys/tree.h>
#include <err.h>
#include <stdio.h>
#include <stdlib.h>

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

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

int
sumaug(struct node *e)
{
	e->sum = e->i;
	if (RB_LEFT(e, entry) != NULL)
		e->sum += RB_LEFT(e, entry)->sum;
	if (RB_RIGHT(e, entry) != NULL)
		e->sum += RB_RIGHT(e, entry)->sum;
}
#define RB_AUGMENT(entry) sumaug(entry)

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\en", n->i);
	}
	print_tree(RB_ROOT(&head));
	printf("Sum of values = %d\n", RB_ROOT(&head)->sum);
	printf("\en");
	return (0);
}
.Ed
.Sh NOTES
Trying to free a tree in the following way is a common error:
.Bd -literal -offset indent
SPLAY_FOREACH(var, NAME, head) {
	SPLAY_REMOVE(NAME, head, var);
	free(var);
}
free(head);
.Ed
.Pp
Since
.Va var
is freed, the
.Fn FOREACH
macro refers to a pointer that may have been reallocated already.
Proper code needs a second variable.
.Bd -literal -offset indent
for (var = SPLAY_MIN(NAME, head); var != NULL; var = nxt) {
	nxt = SPLAY_NEXT(NAME, head, var);
	SPLAY_REMOVE(NAME, head, var);
	free(var);
}
.Ed
.Pp
Both
.Fn RB_INSERT
and
.Fn SPLAY_INSERT
return
.Dv NULL
if the element was inserted in the tree successfully, otherwise they
return a pointer to the element with the colliding key.
.Pp
Accordingly,
.Fn RB_REMOVE
and
.Fn SPLAY_REMOVE
return the pointer to the removed element otherwise they return
.Dv NULL
to indicate an error.
.Sh SEE ALSO
.Xr arb 3 ,
.Xr queue 3
.Rs
.%A "Bernhard Haeupler"
.%A "Siddhartha Sen"
.%A "Robert E. Tarjan"
.%T "Rank-Balanced Trees"
.%U "http://sidsen.azurewebsites.net/papers/rb-trees-talg.pdf"
.%J "ACM Transactions on Algorithms"
.%V "11"
.%N "4"
.%D "June 2015"
.Re
.Sh HISTORY
The tree macros first appeared in
.Fx 4.6 .
.Sh AUTHORS
The author of the tree macros is
.An Niels Provos .