lib/rbtree.c

   1 /*
   2   Red Black Trees
   3   (C) 1999  Andrea Arcangeli <andrea@suse.de>
   4   (C) 2002  David Woodhouse <dwmw2@infradead.org>
   5   (C) 2012  Michel Lespinasse <walken@google.com>
   6
   7   This program is free software; you can redistribute it and/or modify
   8   it under the terms of the GNU General Public License as published by
   9   the Free Software Foundation; either version 2 of the License, or
  10   (at your option) any later version.
  11
  12   This program is distributed in the hope that it will be useful,
  13   but WITHOUT ANY WARRANTY; without even the implied warranty of
  14   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  15   GNU General Public License for more details.
  16
  17   You should have received a copy of the GNU General Public License
  18   along with this program; if not, write to the Free Software
  19   Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  20
  21   linux/lib/rbtree.c
  22 */
  23
  24 #include <linux/rbtree_augmented.h>
  25 #include <linux/export.h>
  26
  27 /*
  28  * red-black trees properties:  http://en.wikipedia.org/wiki/Rbtree
  29  *
  30  *  1) A node is either red or black
  31  *  2) The root is black
  32  *  3) All leaves (NULL) are black
  33  *  4) Both children of every red node are black
  34  *  5) Every simple path from root to leaves contains the same number
  35  *     of black nodes.
  36  *
  37  *  4 and 5 give the O(log n) guarantee, since 4 implies you cannot have two
  38  *  consecutive red nodes in a path and every red node is therefore followed by
  39  *  a black. So if B is the number of black nodes on every simple path (as per
  40  *  5), then the longest possible path due to 4 is 2B.
  41  *
  42  *  We shall indicate color with case, where black nodes are uppercase and red
  43  *  nodes will be lowercase. Unknown color nodes shall be drawn as red within
  44  *  parentheses and have some accompanying text comment.
  45  */
  46
  47 /*
  48  * Notes on lockless lookups:
  49  *
  50  * All stores to the tree structure (rb_left and rb_right) must be done using
  51  * WRITE_ONCE(). And we must not inadvertently cause (temporary) loops in the
  52  * tree structure as seen in program order.
  53  *
  54  * These two requirements will allow lockless iteration of the tree -- not
  55  * correct iteration mind you, tree rotations are not atomic so a lookup might
  56  * miss entire subtrees.
  57  *
  58  * But they do guarantee that any such traversal will only see valid elements
  59  * and that it will indeed complete -- does not get stuck in a loop.
  60  *
  61  * It also guarantees that if the lookup returns an element it is the 'correct'
  62  * one. But not returning an element does _NOT_ mean it's not present.
  63  *
  64  * NOTE:
  65  *
  66  * Stores to __rb_parent_color are not important for simple lookups so those
  67  * are left undone as of now. Nor did I check for loops involving parent
  68  * pointers.
  69  */
  70
  71 static inline void rb_set_black(struct rb_node *rb)
  72 {
  73         rb->__rb_parent_color |= RB_BLACK;
  74 }
  75
  76 static inline struct rb_node *rb_red_parent(struct rb_node *red)
  77 {
  78         return (struct rb_node *)red->__rb_parent_color;
  79 }
  80
  81 /*
  82  * Helper function for rotations:
  83  * - old's parent and color get assigned to new
  84  * - old gets assigned new as a parent and 'color' as a color.
  85  */
  86 static inline void
  87 __rb_rotate_set_parents(struct rb_node *old, struct rb_node *new,
  88                         struct rb_root *root, int color)
  89 {
  90         struct rb_node *parent = rb_parent(old);
  91         new->__rb_parent_color = old->__rb_parent_color;
  92         rb_set_parent_color(old, new, color);
  93         __rb_change_child(old, new, parent, root);
  94 }
  95
  96 static __always_inline void
  97 __rb_insert(struct rb_node *node, struct rb_root *root,
  98             bool newleft, struct rb_node **leftmost,
  99             void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
 100 {
 101         struct rb_node *parent = rb_red_parent(node), *gparent, *tmp;
 102
 103         if (newleft)
 104                 *leftmost = node;
 105
 106         while (true) {
 107                 /*
 108                  * Loop invariant: node is red
 109                  *
 110                  * If there is a black parent, we are done.
 111                  * Otherwise, take some corrective action as we don't
 112                  * want a red root or two consecutive red nodes.
 113                  */
 114                 if (!parent) {
 115                         rb_set_parent_color(node, NULL, RB_BLACK);
 116                         break;
 117                 } else if (rb_is_black(parent))
 118                         break;
 119
 120                 gparent = rb_red_parent(parent);
 121
 122                 tmp = gparent->rb_right;
 123                 if (parent != tmp) {    /* parent == gparent->rb_left */
 124                         if (tmp && rb_is_red(tmp)) {
 125                                 /*
 126                                  * Case 1 - color flips
 127                                  *
 128                                  *       G            g
 129                                  *      / \          / \
 130                                  *     p   u  -->   P   U
 131                                  *    /            /
 132                                  *   n            n
 133                                  *
 134                                  * However, since g's parent might be red, and
 135                                  * 4) does not allow this, we need to recurse
 136                                  * at g.
 137                                  */
 138                                 rb_set_parent_color(tmp, gparent, RB_BLACK);
 139                                 rb_set_parent_color(parent, gparent, RB_BLACK);
 140                                 node = gparent;
 141                                 parent = rb_parent(node);
 142                                 rb_set_parent_color(node, parent, RB_RED);
 143                                 continue;
 144                         }
 145
 146                         tmp = parent->rb_right;
 147                         if (node == tmp) {
 148                                 /*
 149                                  * Case 2 - left rotate at parent
 150                                  *
 151                                  *      G             G
 152                                  *     / \           / \
 153                                  *    p   U  -->    n   U
 154                                  *     \           /
 155                                  *      n         p
 156                                  *
 157                                  * This still leaves us in violation of 4), the
 158                                  * continuation into Case 3 will fix that.
 159                                  */
 160                                 tmp = node->rb_left;
 161                                 WRITE_ONCE(parent->rb_right, tmp);
 162                                 WRITE_ONCE(node->rb_left, parent);
 163                                 if (tmp)
 164                                         rb_set_parent_color(tmp, parent,
 165                                                             RB_BLACK);
 166                                 rb_set_parent_color(parent, node, RB_RED);
 167                                 augment_rotate(parent, node);
 168                                 parent = node;
 169                                 tmp = node->rb_right;
 170                         }
 171
 172                         /*
 173                          * Case 3 - right rotate at gparent
 174                          *
 175                          *        G           P
 176                          *       / \         / \
 177                          *      p   U  -->  n   g
 178                          *     /                 \
 179                          *    n                   U
 180                          */
 181                         WRITE_ONCE(gparent->rb_left, tmp); /* == parent->rb_right */
 182                         WRITE_ONCE(parent->rb_right, gparent);
 183                         if (tmp)
 184                                 rb_set_parent_color(tmp, gparent, RB_BLACK);
 185                         __rb_rotate_set_parents(gparent, parent, root, RB_RED);
 186                         augment_rotate(gparent, parent);
 187                         break;
 188                 } else {
 189                         tmp = gparent->rb_left;
 190                         if (tmp && rb_is_red(tmp)) {
 191                                 /* Case 1 - color flips */
 192                                 rb_set_parent_color(tmp, gparent, RB_BLACK);
 193                                 rb_set_parent_color(parent, gparent, RB_BLACK);
 194                                 node = gparent;
 195                                 parent = rb_parent(node);
 196                                 rb_set_parent_color(node, parent, RB_RED);
 197                                 continue;
 198                         }
 199
 200                         tmp = parent->rb_left;
 201                         if (node == tmp) {
 202                                 /* Case 2 - right rotate at parent */
 203                                 tmp = node->rb_right;
 204                                 WRITE_ONCE(parent->rb_left, tmp);
 205                                 WRITE_ONCE(node->rb_right, parent);
 206                                 if (tmp)
 207                                         rb_set_parent_color(tmp, parent,
 208                                                             RB_BLACK);
 209                                 rb_set_parent_color(parent, node, RB_RED);
 210                                 augment_rotate(parent, node);
 211                                 parent = node;
 212                                 tmp = node->rb_left;
 213                         }
 214
 215                         /* Case 3 - left rotate at gparent */
 216                         WRITE_ONCE(gparent->rb_right, tmp); /* == parent->rb_left */
 217                         WRITE_ONCE(parent->rb_left, gparent);
 218                         if (tmp)
 219                                 rb_set_parent_color(tmp, gparent, RB_BLACK);
 220                         __rb_rotate_set_parents(gparent, parent, root, RB_RED);
 221                         augment_rotate(gparent, parent);
 222                         break;
 223                 }
 224         }
 225 }
 226
 227 /*
 228  * Inline version for rb_erase() use - we want to be able to inline
 229  * and eliminate the dummy_rotate callback there
 230  */
 231 static __always_inline void
 232 ____rb_erase_color(struct rb_node *parent, struct rb_root *root,
 233         void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
 234 {
 235         struct rb_node *node = NULL, *sibling, *tmp1, *tmp2;
 236
 237         while (true) {
 238                 /*
 239                  * Loop invariants:
 240                  * - node is black (or NULL on first iteration)
 241                  * - node is not the root (parent is not NULL)
 242                  * - All leaf paths going through parent and node have a
 243                  *   black node count that is 1 lower than other leaf paths.
 244                  */
 245                 sibling = parent->rb_right;
 246                 if (node != sibling) {  /* node == parent->rb_left */
 247                         if (rb_is_red(sibling)) {
 248                                 /*
 249                                  * Case 1 - left rotate at parent
 250                                  *
 251                                  *     P               S
 252                                  *    / \             / \
 253                                  *   N   s    -->    p   Sr
 254                                  *      / \         / \
 255                                  *     Sl  Sr      N   Sl
 256                                  */
 257                                 tmp1 = sibling->rb_left;
 258                                 WRITE_ONCE(parent->rb_right, tmp1);
 259                                 WRITE_ONCE(sibling->rb_left, parent);
 260                                 rb_set_parent_color(tmp1, parent, RB_BLACK);
 261                                 __rb_rotate_set_parents(parent, sibling, root,
 262                                                         RB_RED);
 263                                 augment_rotate(parent, sibling);
 264                                 sibling = tmp1;
 265                         }
 266                         tmp1 = sibling->rb_right;
 267                         if (!tmp1 || rb_is_black(tmp1)) {
 268                                 tmp2 = sibling->rb_left;
 269                                 if (!tmp2 || rb_is_black(tmp2)) {
 270                                         /*
 271                                          * Case 2 - sibling color flip
 272                                          * (p could be either color here)
 273                                          *
 274                                          *    (p)           (p)
 275                                          *    / \           / \
 276                                          *   N   S    -->  N   s
 277                                          *      / \           / \
 278                                          *     Sl  Sr        Sl  Sr
 279                                          *
 280                                          * This leaves us violating 5) which
 281                                          * can be fixed by flipping p to black
 282                                          * if it was red, or by recursing at p.
 283                                          * p is red when coming from Case 1.
 284                                          */
 285                                         rb_set_parent_color(sibling, parent,
 286                                                             RB_RED);
 287                                         if (rb_is_red(parent))
 288                                                 rb_set_black(parent);
 289                                         else {
 290                                                 node = parent;
 291                                                 parent = rb_parent(node);
 292                                                 if (parent)
 293                                                         continue;
 294                                         }
 295                                         break;
 296                                 }
 297                                 /*
 298                                  * Case 3 - right rotate at sibling
 299                                  * (p could be either color here)
 300                                  *
 301                                  *   (p)           (p)
 302                                  *   / \           / \
 303                                  *  N   S    -->  N   sl
 304                                  *     / \             \
 305                                  *    sl  Sr            S
 306                                  *                       \
 307                                  *                        Sr
 308                                  *
 309                                  * Note: p might be red, and then both
 310                                  * p and sl are red after rotation(which
 311                                  * breaks property 4). This is fixed in
 312                                  * Case 4 (in __rb_rotate_set_parents()
 313                                  *         which set sl the color of p
 314                                  *         and set p RB_BLACK)
 315                                  *
 316                                  *   (p)            (sl)
 317                                  *   / \            /  \
 318                                  *  N   sl   -->   P    S
 319                                  *       \        /      \
 320                                  *        S      N        Sr
 321                                  *         \
 322                                  *          Sr
 323                                  */
 324                                 tmp1 = tmp2->rb_right;
 325                                 WRITE_ONCE(sibling->rb_left, tmp1);
 326                                 WRITE_ONCE(tmp2->rb_right, sibling);
 327                                 WRITE_ONCE(parent->rb_right, tmp2);
 328                                 if (tmp1)
 329                                         rb_set_parent_color(tmp1, sibling,
 330                                                             RB_BLACK);
 331                                 augment_rotate(sibling, tmp2);
 332                                 tmp1 = sibling;
 333                                 sibling = tmp2;
 334                         }
 335                         /*
 336                          * Case 4 - left rotate at parent + color flips
 337                          * (p and sl could be either color here.
 338                          *  After rotation, p becomes black, s acquires
 339                          *  p's color, and sl keeps its color)
 340                          *
 341                          *      (p)             (s)
 342                          *      / \             / \
 343                          *     N   S     -->   P   Sr
 344                          *        / \         / \
 345                          *      (sl) sr      N  (sl)
 346                          */
 347                         tmp2 = sibling->rb_left;
 348                         WRITE_ONCE(parent->rb_right, tmp2);
 349                         WRITE_ONCE(sibling->rb_left, parent);
 350                         rb_set_parent_color(tmp1, sibling, RB_BLACK);
 351                         if (tmp2)
 352                                 rb_set_parent(tmp2, parent);
 353                         __rb_rotate_set_parents(parent, sibling, root,
 354                                                 RB_BLACK);
 355                         augment_rotate(parent, sibling);
 356                         break;
 357                 } else {
 358                         sibling = parent->rb_left;
 359                         if (rb_is_red(sibling)) {
 360                                 /* Case 1 - right rotate at parent */
 361                                 tmp1 = sibling->rb_right;
 362                                 WRITE_ONCE(parent->rb_left, tmp1);
 363                                 WRITE_ONCE(sibling->rb_right, parent);
 364                                 rb_set_parent_color(tmp1, parent, RB_BLACK);
 365                                 __rb_rotate_set_parents(parent, sibling, root,
 366                                                         RB_RED);
 367                                 augment_rotate(parent, sibling);
 368                                 sibling = tmp1;
 369                         }
 370                         tmp1 = sibling->rb_left;
 371                         if (!tmp1 || rb_is_black(tmp1)) {
 372                                 tmp2 = sibling->rb_right;
 373                                 if (!tmp2 || rb_is_black(tmp2)) {
 374                                         /* Case 2 - sibling color flip */
 375                                         rb_set_parent_color(sibling, parent,
 376                                                             RB_RED);
 377                                         if (rb_is_red(parent))
 378                                                 rb_set_black(parent);
 379                                         else {
 380                                                 node = parent;
 381                                                 parent = rb_parent(node);
 382                                                 if (parent)
 383                                                         continue;
 384                                         }
 385                                         break;
 386                                 }
 387                                 /* Case 3 - left rotate at sibling */
 388                                 tmp1 = tmp2->rb_left;
 389                                 WRITE_ONCE(sibling->rb_right, tmp1);
 390                                 WRITE_ONCE(tmp2->rb_left, sibling);
 391                                 WRITE_ONCE(parent->rb_left, tmp2);
 392                                 if (tmp1)
 393                                         rb_set_parent_color(tmp1, sibling,
 394                                                             RB_BLACK);
 395                                 augment_rotate(sibling, tmp2);
 396                                 tmp1 = sibling;
 397                                 sibling = tmp2;
 398                         }
 399                         /* Case 4 - right rotate at parent + color flips */
 400                         tmp2 = sibling->rb_right;
 401                         WRITE_ONCE(parent->rb_left, tmp2);
 402                         WRITE_ONCE(sibling->rb_right, parent);
 403                         rb_set_parent_color(tmp1, sibling, RB_BLACK);
 404                         if (tmp2)
 405                                 rb_set_parent(tmp2, parent);
 406                         __rb_rotate_set_parents(parent, sibling, root,
 407                                                 RB_BLACK);
 408                         augment_rotate(parent, sibling);
 409                         break;
 410                 }
 411         }
 412 }
 413
 414 /* Non-inline version for rb_erase_augmented() use */
 415 void __rb_erase_color(struct rb_node *parent, struct rb_root *root,
 416         void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
 417 {
 418         ____rb_erase_color(parent, root, augment_rotate);
 419 }
 420 EXPORT_SYMBOL(__rb_erase_color);
 421
 422 /*
 423  * Non-augmented rbtree manipulation functions.
 424  *
 425  * We use dummy augmented callbacks here, and have the compiler optimize them
 426  * out of the rb_insert_color() and rb_erase() function definitions.
 427  */
 428
 429 static inline void dummy_propagate(struct rb_node *node, struct rb_node *stop) {}
 430 static inline void dummy_copy(struct rb_node *old, struct rb_node *new) {}
 431 static inline void dummy_rotate(struct rb_node *old, struct rb_node *new) {}
 432
 433 static const struct rb_augment_callbacks dummy_callbacks = {
 434         .propagate = dummy_propagate,
 435         .copy = dummy_copy,
 436         .rotate = dummy_rotate
 437 };
 438
 439 void rb_insert_color(struct rb_node *node, struct rb_root *root)
 440 {
 441         __rb_insert(node, root, false, NULL, dummy_rotate);
 442 }
 443 EXPORT_SYMBOL(rb_insert_color);
 444
 445 void rb_erase(struct rb_node *node, struct rb_root *root)
 446 {
 447         struct rb_node *rebalance;
 448         rebalance = __rb_erase_augmented(node, root,
 449                                          NULL, &dummy_callbacks);
 450         if (rebalance)
 451                 ____rb_erase_color(rebalance, root, dummy_rotate);
 452 }
 453 EXPORT_SYMBOL(rb_erase);
 454
 455 void rb_insert_color_cached(struct rb_node *node,
 456                             struct rb_root_cached *root, bool leftmost)
 457 {
 458         __rb_insert(node, &root->rb_root, leftmost,
 459                     &root->rb_leftmost, dummy_rotate);
 460 }
 461 EXPORT_SYMBOL(rb_insert_color_cached);
 462
 463 void rb_erase_cached(struct rb_node *node, struct rb_root_cached *root)
 464 {
 465         struct rb_node *rebalance;
 466         rebalance = __rb_erase_augmented(node, &root->rb_root,
 467                                          &root->rb_leftmost, &dummy_callbacks);
 468         if (rebalance)
 469                 ____rb_erase_color(rebalance, &root->rb_root, dummy_rotate);
 470 }
 471 EXPORT_SYMBOL(rb_erase_cached);
 472
 473 /*
 474  * Augmented rbtree manipulation functions.
 475  *
 476  * This instantiates the same __always_inline functions as in the non-augmented
 477  * case, but this time with user-defined callbacks.
 478  */
 479
 480 void __rb_insert_augmented(struct rb_node *node, struct rb_root *root,
 481                            bool newleft, struct rb_node **leftmost,
 482         void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
 483 {
 484         __rb_insert(node, root, newleft, leftmost, augment_rotate);
 485 }
 486 EXPORT_SYMBOL(__rb_insert_augmented);
 487
 488 /*
 489  * This function returns the first node (in sort order) of the tree.
 490  */
 491 struct rb_node *rb_first(const struct rb_root *root)
 492 {
 493         struct rb_node  *n;
 494
 495         n = root->rb_node;
 496         if (!n)
 497                 return NULL;
 498         while (n->rb_left)
 499                 n = n->rb_left;
 500         return n;
 501 }
 502 EXPORT_SYMBOL(rb_first);
 503
 504 struct rb_node *rb_last(const struct rb_root *root)
 505 {
 506         struct rb_node  *n;
 507
 508         n = root->rb_node;
 509         if (!n)
 510                 return NULL;
 511         while (n->rb_right)
 512                 n = n->rb_right;
 513         return n;
 514 }
 515 EXPORT_SYMBOL(rb_last);
 516
 517 struct rb_node *rb_next(const struct rb_node *node)
 518 {
 519         struct rb_node *parent;
 520
 521         if (RB_EMPTY_NODE(node))
 522                 return NULL;
 523
 524         /*
 525          * If we have a right-hand child, go down and then left as far
 526          * as we can.
 527          */
 528         if (node->rb_right) {
 529                 node = node->rb_right;
 530                 while (node->rb_left)
 531                         node=node->rb_left;
 532                 return (struct rb_node *)node;
 533         }
 534
 535         /*
 536          * No right-hand children. Everything down and left is smaller than us,
 537          * so any 'next' node must be in the general direction of our parent.
 538          * Go up the tree; any time the ancestor is a right-hand child of its
 539          * parent, keep going up. First time it's a left-hand child of its
 540          * parent, said parent is our 'next' node.
 541          */
 542         while ((parent = rb_parent(node)) && node == parent->rb_right)
 543                 node = parent;
 544
 545         return parent;
 546 }
 547 EXPORT_SYMBOL(rb_next);
 548
 549 struct rb_node *rb_prev(const struct rb_node *node)
 550 {
 551         struct rb_node *parent;
 552
 553         if (RB_EMPTY_NODE(node))
 554                 return NULL;
 555
 556         /*
 557          * If we have a left-hand child, go down and then right as far
 558          * as we can.
 559          */
 560         if (node->rb_left) {
 561                 node = node->rb_left;
 562                 while (node->rb_right)
 563                         node=node->rb_right;
 564                 return (struct rb_node *)node;
 565         }
 566
 567         /*
 568          * No left-hand children. Go up till we find an ancestor which
 569          * is a right-hand child of its parent.
 570          */
 571         while ((parent = rb_parent(node)) && node == parent->rb_left)
 572                 node = parent;
 573
 574         return parent;
 575 }
 576 EXPORT_SYMBOL(rb_prev);
 577
 578 void rb_replace_node(struct rb_node *victim, struct rb_node *new,
 579                      struct rb_root *root)
 580 {
 581         struct rb_node *parent = rb_parent(victim);
 582
 583         /* Copy the pointers/colour from the victim to the replacement */
 584         *new = *victim;
 585
 586         /* Set the surrounding nodes to point to the replacement */
 587         if (victim->rb_left)
 588                 rb_set_parent(victim->rb_left, new);
 589         if (victim->rb_right)
 590                 rb_set_parent(victim->rb_right, new);
 591         __rb_change_child(victim, new, parent, root);
 592 }
 593 EXPORT_SYMBOL(rb_replace_node);
 594
 595 void rb_replace_node_rcu(struct rb_node *victim, struct rb_node *new,
 596                          struct rb_root *root)
 597 {
 598         struct rb_node *parent = rb_parent(victim);
 599
 600         /* Copy the pointers/colour from the victim to the replacement */
 601         *new = *victim;
 602
 603         /* Set the surrounding nodes to point to the replacement */
 604         if (victim->rb_left)
 605                 rb_set_parent(victim->rb_left, new);
 606         if (victim->rb_right)
 607                 rb_set_parent(victim->rb_right, new);
 608
 609         /* Set the parent's pointer to the new node last after an RCU barrier
 610          * so that the pointers onwards are seen to be set correctly when doing
 611          * an RCU walk over the tree.
 612          */
 613         __rb_change_child_rcu(victim, new, parent, root);
 614 }
 615 EXPORT_SYMBOL(rb_replace_node_rcu);
 616
 617 static struct rb_node *rb_left_deepest_node(const struct rb_node *node)
 618 {
 619         for (;;) {
 620                 if (node->rb_left)
 621                         node = node->rb_left;
 622                 else if (node->rb_right)
 623                         node = node->rb_right;
 624                 else
 625                         return (struct rb_node *)node;
 626         }
 627 }
 628
 629 struct rb_node *rb_next_postorder(const struct rb_node *node)
 630 {
 631         const struct rb_node *parent;
 632         if (!node)
 633                 return NULL;
 634         parent = rb_parent(node);
 635
 636         /* If we're sitting on node, we've already seen our children */
 637         if (parent && node == parent->rb_left && parent->rb_right) {
 638                 /* If we are the parent's left node, go to the parent's right
 639                  * node then all the way down to the left */
 640                 return rb_left_deepest_node(parent->rb_right);
 641         } else
 642                 /* Otherwise we are the parent's right node, and the parent
 643                  * should be next */
 644                 return (struct rb_node *)parent;
 645 }
 646 EXPORT_SYMBOL(rb_next_postorder);
 647
 648 struct rb_node *rb_first_postorder(const struct rb_root *root)
 649 {
 650         if (!root->rb_node)
 651                 return NULL;
 652
 653         return rb_left_deepest_node(root->rb_node);
 654 }
 655 EXPORT_SYMBOL(rb_first_postorder);