diff --git a/share/man/man3/tree.3 b/share/man/man3/tree.3 index cbc741fe642d..08d442db5e78 100644 --- a/share/man/man3/tree.3 +++ b/share/man/man3/tree.3 @@ -1,764 +1,766 @@ .\" $OpenBSD: tree.3,v 1.7 2002/06/12 01:09:20 provos Exp $ .\" .\" Copyright 2002 Niels Provos .\" All rights reserved. .\" .\" Redistribution and use in source and binary forms, with or without .\" modification, are permitted provided that the following conditions .\" are met: .\" 1. Redistributions of source code must retain the above copyright .\" notice, this list of conditions and the following disclaimer. .\" 2. Redistributions in binary form must reproduce the above copyright .\" notice, this list of conditions and the following disclaimer in the .\" documentation and/or other materials provided with the distribution. .\" 3. All advertising materials mentioning features or use of this software .\" must display the following acknowledgement: .\" This product includes software developed by Niels Provos. .\" 4. The name of the author may not be used to endorse or promote products .\" derived from this software without specific prior written permission. .\" .\" THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR .\" IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES .\" OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. .\" IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, .\" INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT .\" NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, .\" DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY .\" THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT .\" (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF .\" THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. .\" .\" $FreeBSD$ .\" .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 +If +.Fn 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 #include #include #include 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 .