common/avl/avl.c

/*
 * CDDL HEADER START
 *
 * The contents of this file are subject to the terms of the
 * Common Development and Distribution License, Version 1.0 only
 * (the "License").  You may not use this file except in compliance
 * with the License.
 *
 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
 * or http://www.opensolaris.org/os/licensing.
 * See the License for the specific language governing permissions
 * and limitations under the License.
 *
 * When distributing Covered Code, include this CDDL HEADER in each
 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
 * If applicable, add the following below this CDDL HEADER, with the
 * fields enclosed by brackets "[]" replaced with your own identifying
 * information: Portions Copyright [yyyy] [name of copyright owner]
 *
 * CDDL HEADER END
 */
/*
 * Copyright 2004 Sun Microsystems, Inc.  All rights reserved.
 * Use is subject to license terms.
 */

#pragma ident	"%Z%%M%	%I%	%E% SMI"


/*
 * AVL - generic AVL tree implementation for kernel use
 *
 * A complete description of AVL trees can be found in many CS textbooks.
 *
 * Here is a very brief overview. An AVL tree is a binary search tree that is
 * almost perfectly balanced. By "almost" perfectly balanced, we mean that at
 * any given node, the left and right subtrees are allowed to differ in height
 * by at most 1 level.
 *
 * This relaxation from a perfectly balanced binary tree allows doing
 * insertion and deletion relatively efficiently. Searching the tree is
 * still a fast operation, roughly O(log(N)).
 *
 * The key to insertion and deletion is a set of tree maniuplations called
 * rotations, which bring unbalanced subtrees back into the semi-balanced state.
 *
 * This implementation of AVL trees has the following peculiarities:
 *
 *	- The AVL specific data structures are physically embedded as fields
 *	  in the "using" data structures.  To maintain generality the code
 *	  must constantly translate between "avl_node_t *" and containing
 *	  data structure "void *"s by adding/subracting the avl_offset.
 *
 *	- Since the AVL data is always embedded in other structures, there is
 *	  no locking or memory allocation in the AVL routines. This must be
 *	  provided for by the enclosing data structure's semantics. Typically,
 *	  avl_insert()/_remove()/avl_insert_here() require some kind of
 *	  exclusive write lock. Other operations require a read lock.
 *
 *      - The implementation uses iteration instead of explicit recursion,
 *	  since it is intended to run on limited size kernel stacks. Since
 *	  there is no recursion stack present to move "up" in the tree,
 *	  there is an explicit "parent" link in the avl_node_t.
 *
 *      - The left/right children pointers of a node are in an array.
 *	  In the code, variables (instead of constants) are used to represent
 *	  left and right indices.  The implementation is written as if it only
 *	  dealt with left handed manipulations.  By changing the value assigned
 *	  to "left", the code also works for right handed trees.  The
 *	  following variables/terms are frequently used:
 *
 *		int left;	// 0 when dealing with left children,
 *				// 1 for dealing with right children
 *
 *		int left_heavy;	// -1 when left subtree is taller at some node,
 *				// +1 when right subtree is taller
 *
 *		int right;	// will be the opposite of left (0 or 1)
 *		int right_heavy;// will be the opposite of left_heavy (-1 or 1)
 *
 *		int direction;  // 0 for "<" (ie. left child); 1 for ">" (right)
 *
 *	  Though it is a little more confusing to read the code, the approach
 *	  allows using half as much code (and hence cache footprint) for tree
 *	  manipulations and eliminates many conditional branches.
 *
 *	- The avl_index_t is an opaque "cookie" used to find nodes at or
 *	  adjacent to where a new value would be inserted in the tree. The value
 *	  is a modified "avl_node_t *".  The bottom bit (normally 0 for a
 *	  pointer) is set to indicate if that the new node has a value greater
 *	  than the value of the indicated "avl_node_t *".
 */

#include <sys/types.h>
#include <sys/param.h>
#include <sys/debug.h>
#include <sys/avl.h>

/*
 * Small arrays to translate between balance (or diff) values and child indeces.
 *
 * Code that deals with binary tree data structures will randomly use
 * left and right children when examining a tree.  C "if()" statements
 * which evaluate randomly suffer from very poor hardware branch prediction.
 * In this code we avoid some of the branch mispredictions by using the
 * following translation arrays. They replace random branches with an
 * additional memory reference. Since the translation arrays are both very
 * small the data should remain efficiently in cache.
 */
static const int  avl_child2balance[2]	= {-1, 1};
static const int  avl_balance2child[]	= {0, 0, 1};


/*
 * Walk from one node to the previous valued node (ie. an infix walk
 * towards the left). At any given node we do one of 2 things:
 *
 * - If there is a left child, go to it, then to it's rightmost descendant.
 *
 * - otherwise we return thru parent nodes until we've come from a right child.
 *
 * Return Value:
 * NULL - if at the end of the nodes
 * otherwise next node
 */
void *
avl_walk(avl_tree_t *tree, void	*oldnode, int left)
{
	size_t off = tree->avl_offset;
	avl_node_t *node = AVL_DATA2NODE(oldnode, off);
	int right = 1 - left;
	int was_child;


	/*
	 * nowhere to walk to if tree is empty
	 */
	if (node == NULL)
		return (NULL);

	/*
	 * Visit the previous valued node. There are two possibilities:
	 *
	 * If this node has a left child, go down one left, then all
	 * the way right.
	 */
	if (node->avl_child[left] != NULL) {
		for (node = node->avl_child[left];
		    node->avl_child[right] != NULL;
		    node = node->avl_child[right])
			;
	/*
	 * Otherwise, return thru left children as far as we can.
	 */
	} else {
		for (;;) {
			was_child = AVL_XCHILD(node);
			node = AVL_XPARENT(node);
			if (node == NULL)
				return (NULL);
			if (was_child == right)
				break;
		}
	}

	return (AVL_NODE2DATA(node, off));
}

/*
 * Return the lowest valued node in a tree or NULL.
 * (leftmost child from root of tree)
 */
void *
avl_first(avl_tree_t *tree)
{
	avl_node_t *node;
	avl_node_t *prev = NULL;
	size_t off = tree->avl_offset;

	for (node = tree->avl_root; node != NULL; node = node->avl_child[0])
		prev = node;

	if (prev != NULL)
		return (AVL_NODE2DATA(prev, off));
	return (NULL);
}

/*
 * Return the highest valued node in a tree or NULL.
 * (rightmost child from root of tree)
 */
void *
avl_last(avl_tree_t *tree)
{
	avl_node_t *node;
	avl_node_t *prev = NULL;
	size_t off = tree->avl_offset;

	for (node = tree->avl_root; node != NULL; node = node->avl_child[1])
		prev = node;

	if (prev != NULL)
		return (AVL_NODE2DATA(prev, off));
	return (NULL);
}

/*
 * Access the node immediately before or after an insertion point.
 *
 * "avl_index_t" is a (avl_node_t *) with the bottom bit indicating a child
 *
 * Return value:
 *	NULL: no node in the given direction
 *	"void *"  of the found tree node
 */
void *
avl_nearest(avl_tree_t *tree, avl_index_t where, int direction)
{
	int child = AVL_INDEX2CHILD(where);
	avl_node_t *node = AVL_INDEX2NODE(where);
	void *data;
	size_t off = tree->avl_offset;

	if (node == NULL) {
		ASSERT(tree->avl_root == NULL);
		return (NULL);
	}
	data = AVL_NODE2DATA(node, off);
	if (child != direction)
		return (data);

	return (avl_walk(tree, data, direction));
}


/*
 * Search for the node which contains "value".  The algorithm is a
 * simple binary tree search.
 *
 * return value:
 *	NULL: the value is not in the AVL tree
 *		*where (if not NULL)  is set to indicate the insertion point
 *	"void *"  of the found tree node
 */
void *
avl_find(avl_tree_t *tree, void *value, avl_index_t *where)
{
	avl_node_t *node;
	avl_node_t *prev = NULL;
	int child = 0;
	int diff;
	size_t off = tree->avl_offset;

	for (node = tree->avl_root; node != NULL;
	    node = node->avl_child[child]) {

		prev = node;

		diff = tree->avl_compar(value, AVL_NODE2DATA(node, off));
		ASSERT(-1 <= diff && diff <= 1);
		if (diff == 0) {
#ifdef DEBUG
			if (where != NULL)
				*where = NULL;
#endif
			return (AVL_NODE2DATA(node, off));
		}
		child = avl_balance2child[1 + diff];

	}

	if (where != NULL)
		*where = AVL_MKINDEX(prev, child);

	return (NULL);
}


/*
 * Perform a rotation to restore balance at the subtree given by depth.
 *
 * This routine is used by both insertion and deletion. The return value
 * indicates:
 *	 0 : subtree did not change height
 *	!0 : subtree was reduced in height
 *
 * The code is written as if handling left rotations, right rotations are
 * symmetric and handled by swapping values of variables right/left[_heavy]
 *
 * On input balance is the "new" balance at "node". This value is either
 * -2 or +2.
 */
static int
avl_rotation(avl_tree_t *tree, avl_node_t *node, int balance)
{
	int left = !(balance < 0);	/* when balance = -2, left will be 0 */
	int right = 1 - left;
	int left_heavy = balance >> 1;
	int right_heavy = -left_heavy;
	avl_node_t *parent = AVL_XPARENT(node);
	avl_node_t *child = node->avl_child[left];
	avl_node_t *cright;
	avl_node_t *gchild;
	avl_node_t *gright;
	avl_node_t *gleft;
	int which_child = AVL_XCHILD(node);
	int child_bal = AVL_XBALANCE(child);

	/* BEGIN CSTYLED */
	/*
	 * case 1 : node is overly left heavy, the left child is balanced or
	 * also left heavy. This requires the following rotation.
	 *
	 *                   (node bal:-2)
	 *                    /           \
	 *                   /             \
	 *              (child bal:0 or -1)
	 *              /    \
	 *             /      \
	 *                     cright
	 *
	 * becomes:
	 *
	 *              (child bal:1 or 0)
	 *              /        \
	 *             /          \
	 *                        (node bal:-1 or 0)
	 *                         /     \
	 *                        /       \
	 *                     cright
	 *
	 * we detect this situation by noting that child's balance is not
	 * right_heavy.
	 */
	/* END CSTYLED */
	if (child_bal != right_heavy) {

		/*
		 * compute new balance of nodes
		 *
		 * If child used to be left heavy (now balanced) we reduced
		 * the height of this sub-tree -- used in "return...;" below
		 */
		child_bal += right_heavy; /* adjust towards right */

		/*
		 * move "cright" to be node's left child
		 */
		cright = child->avl_child[right];
		node->avl_child[left] = cright;
		if (cright != NULL) {
			AVL_SETPARENT(cright, node);
			AVL_SETCHILD(cright, left);
		}

		/*
		 * move node to be child's right child
		 */
		child->avl_child[right] = node;
		AVL_SETBALANCE(node, -child_bal);
		AVL_SETCHILD(node, right);
		AVL_SETPARENT(node, child);

		/*
		 * update the pointer into this subtree
		 */
		AVL_SETBALANCE(child, child_bal);
		AVL_SETCHILD(child, which_child);
		AVL_SETPARENT(child, parent);
		if (parent != NULL)
			parent->avl_child[which_child] = child;
		else
			tree->avl_root = child;

		return (child_bal == 0);
	}

	/* BEGIN CSTYLED */
	/*
	 * case 2 : When node is left heavy, but child is right heavy we use
	 * a different rotation.
	 *
	 *                   (node b:-2)
	 *                    /   \
	 *                   /     \
	 *                  /       \
	 *             (child b:+1)
	 *              /     \
	 *             /       \
	 *                   (gchild b: != 0)
	 *                     /  \
	 *                    /    \
	 *                 gleft   gright
	 *
	 * becomes:
	 *
	 *              (gchild b:0)
	 *              /       \
	 *             /         \
	 *            /           \
	 *        (child b:?)   (node b:?)
	 *         /  \          /   \
	 *        /    \        /     \
	 *            gleft   gright
	 *
	 * computing the new balances is more complicated. As an example:
	 *	 if gchild was right_heavy, then child is now left heavy
	 *		else it is balanced
	 */
	/* END CSTYLED */
	gchild = child->avl_child[right];
	gleft = gchild->avl_child[left];
	gright = gchild->avl_child[right];

	/*
	 * move gright to left child of node and
	 *
	 * move gleft to right child of node
	 */
	node->avl_child[left] = gright;
	if (gright != NULL) {
		AVL_SETPARENT(gright, node);
		AVL_SETCHILD(gright, left);
	}

	child->avl_child[right] = gleft;
	if (gleft != NULL) {
		AVL_SETPARENT(gleft, child);
		AVL_SETCHILD(gleft, right);
	}

	/*
	 * move child to left child of gchild and
	 *
	 * move node to right child of gchild and
	 *
	 * fixup parent of all this to point to gchild
	 */
	balance = AVL_XBALANCE(gchild);
	gchild->avl_child[left] = child;
	AVL_SETBALANCE(child, (balance == right_heavy ? left_heavy : 0));
	AVL_SETPARENT(child, gchild);
	AVL_SETCHILD(child, left);

	gchild->avl_child[right] = node;
	AVL_SETBALANCE(node, (balance == left_heavy ? right_heavy : 0));
	AVL_SETPARENT(node, gchild);
	AVL_SETCHILD(node, right);

	AVL_SETBALANCE(gchild, 0);
	AVL_SETPARENT(gchild, parent);
	AVL_SETCHILD(gchild, which_child);
	if (parent != NULL)
		parent->avl_child[which_child] = gchild;
	else
		tree->avl_root = gchild;

	return (1);	/* the new tree is always shorter */
}


/*
 * Insert a new node into an AVL tree at the specified (from avl_find()) place.
 *
 * Newly inserted nodes are always leaf nodes in the tree, since avl_find()
 * searches out to the leaf positions.  The avl_index_t indicates the node
 * which will be the parent of the new node.
 *
 * After the node is inserted, a single rotation further up the tree may
 * be necessary to maintain an acceptable AVL balance.
 */
void
avl_insert(avl_tree_t *tree, void *new_data, avl_index_t where)
{
	avl_node_t *node;
	avl_node_t *parent = AVL_INDEX2NODE(where);
	int old_balance;
	int new_balance;
	int which_child = AVL_INDEX2CHILD(where);
	size_t off = tree->avl_offset;

	ASSERT(tree);
#ifdef _LP64
	ASSERT(((uintptr_t)new_data & 0x7) == 0);
#endif

	node = AVL_DATA2NODE(new_data, off);

	/*
	 * First, add the node to the tree at the indicated position.
	 */
	++tree->avl_numnodes;

	node->avl_child[0] = NULL;
	node->avl_child[1] = NULL;

	AVL_SETCHILD(node, which_child);
	AVL_SETBALANCE(node, 0);
	AVL_SETPARENT(node, parent);
	if (parent != NULL) {
		ASSERT(parent->avl_child[which_child] == NULL);
		parent->avl_child[which_child] = node;
	} else {
		ASSERT(tree->avl_root == NULL);
		tree->avl_root = node;
	}
	/*
	 * Now, back up the tree modifying the balance of all nodes above the
	 * insertion point. If we get to a highly unbalanced ancestor, we
	 * need to do a rotation.  If we back out of the tree we are done.
	 * If we brought any subtree into perfect balance (0), we are also done.
	 */
	for (;;) {
		node = parent;
		if (node == NULL)
			return;

		/*
		 * Compute the new balance
		 */
		old_balance = AVL_XBALANCE(node);
		new_balance = old_balance + avl_child2balance[which_child];

		/*
		 * If we introduced equal balance, then we are done immediately
		 */
		if (new_balance == 0) {
			AVL_SETBALANCE(node, 0);
			return;
		}

		/*
		 * If both old and new are not zero we went
		 * from -1 to -2 balance, do a rotation.
		 */
		if (old_balance != 0)
			break;

		AVL_SETBALANCE(node, new_balance);
		parent = AVL_XPARENT(node);
		which_child = AVL_XCHILD(node);
	}

	/*
	 * perform a rotation to fix the tree and return
	 */
	(void) avl_rotation(tree, node, new_balance);
}

/*
 * Insert "new_data" in "tree" in the given "direction" either after or
 * before (AVL_AFTER, AVL_BEFORE) the data "here".
 *
 * Insertions can only be done at empty leaf points in the tree, therefore
 * if the given child of the node is already present we move to either
 * the AVL_PREV or AVL_NEXT and reverse the insertion direction. Since
 * every other node in the tree is a leaf, this always works.
 *
 * To help developers using this interface, we assert that the new node
 * is correctly ordered at every step of the way in DEBUG kernels.
 */
void
avl_insert_here(
	avl_tree_t *tree,
	void *new_data,
	void *here,
	int direction)
{
	avl_node_t *node;
	int child = direction;	/* rely on AVL_BEFORE == 0, AVL_AFTER == 1 */

	ASSERT(tree != NULL);
	ASSERT(new_data != NULL);
	ASSERT(here != NULL);
	ASSERT(direction == AVL_BEFORE || direction == AVL_AFTER);

	/*
	 * If corresponding child of node is not NULL, go to the neighboring
	 * node and reverse the insertion direction.
	 */
	node = AVL_DATA2NODE(here, tree->avl_offset);
	ASSERT(tree->avl_compar(new_data, here) > 0 ? child == 1 : child == 0);

	if (node->avl_child[child] != NULL) {
		node = node->avl_child[child];
		child = 1 - child;
		while (node->avl_child[child] != NULL) {
			ASSERT(tree->avl_compar(new_data,
			    AVL_NODE2DATA(node, tree->avl_offset)) > 0 ?
			    child == 1 : child == 0);
			node = node->avl_child[child];
		}
		ASSERT(tree->avl_compar(new_data,
		    AVL_NODE2DATA(node, tree->avl_offset)) > 0 ?
		    child == 1 : child == 0);
	}
	ASSERT(node->avl_child[child] == NULL);

	avl_insert(tree, new_data, AVL_MKINDEX(node, child));
}

/*
 * Delete a node from the AVL tree.  Deletion is similar to insertion, but
 * with 2 complications.
 *
 * First, we may be deleting an interior node. Consider the following subtree:
 *
 *     d           c            c
 *    / \         / \          / \
 *   b   e       b   e        b   e
 *  / \	        / \          /
 * a   c       a            a
 *
 * When we are deleting node (d), we find and bring up an adjacent valued leaf
 * node, say (c), to take the interior node's place. In the code this is
 * handled by temporarily swapping (d) and (c) in the tree and then using
 * common code to delete (d) from the leaf position.
 *
 * Secondly, an interior deletion from a deep tree may require more than one
 * rotation to fix the balance. This is handled by moving up the tree through
 * parents and applying rotations as needed. The return value from
 * avl_rotation() is used to detect when a subtree did not change overall
 * height due to a rotation.
 */
void
avl_remove(avl_tree_t *tree, void *data)
{
	avl_node_t *delete;
	avl_node_t *parent;
	avl_node_t *node;
	avl_node_t tmp;
	int old_balance;
	int new_balance;
	int left;
	int right;
	int which_child;
	size_t off = tree->avl_offset;

	ASSERT(tree);

	delete = AVL_DATA2NODE(data, off);

	/*
	 * Deletion is easiest with a node that has at most 1 child.
	 * We swap a node with 2 children with a sequentially valued
	 * neighbor node. That node will have at most 1 child. Note this
	 * has no effect on the ordering of the remaining nodes.
	 *
	 * As an optimization, we choose the greater neighbor if the tree
	 * is right heavy, otherwise the left neighbor. This reduces the
	 * number of rotations needed.
	 */
	if (delete->avl_child[0] != NULL && delete->avl_child[1] != NULL) {

		/*
		 * choose node to swap from whichever side is taller
		 */
		old_balance = AVL_XBALANCE(delete);
		left = avl_balance2child[old_balance + 1];
		right = 1 - left;

		/*
		 * get to the previous value'd node
		 * (down 1 left, as far as possible right)
		 */
		for (node = delete->avl_child[left];
		    node->avl_child[right] != NULL;
		    node = node->avl_child[right])
			;

		/*
		 * create a temp placeholder for 'node'
		 * move 'node' to delete's spot in the tree
		 */
		tmp = *node;

		*node = *delete;
		if (node->avl_child[left] == node)
			node->avl_child[left] = &tmp;

		parent = AVL_XPARENT(node);
		if (parent != NULL)
			parent->avl_child[AVL_XCHILD(node)] = node;
		else
			tree->avl_root = node;
		AVL_SETPARENT(node->avl_child[left], node);
		AVL_SETPARENT(node->avl_child[right], node);

		/*
		 * Put tmp where node used to be (just temporary).
		 * It always has a parent and at most 1 child.
		 */
		delete = &tmp;
		parent = AVL_XPARENT(delete);
		parent->avl_child[AVL_XCHILD(delete)] = delete;
		which_child = (delete->avl_child[1] != 0);
		if (delete->avl_child[which_child] != NULL)
			AVL_SETPARENT(delete->avl_child[which_child], delete);
	}


	/*
	 * Here we know "delete" is at least partially a leaf node. It can
	 * be easily removed from the tree.
	 */
	--tree->avl_numnodes;
	parent = AVL_XPARENT(delete);
	which_child = AVL_XCHILD(delete);
	if (delete->avl_child[0] != NULL)
		node = delete->avl_child[0];
	else
		node = delete->avl_child[1];

	/*
	 * Connect parent directly to node (leaving out delete).
	 */
	if (node != NULL) {
		AVL_SETPARENT(node, parent);
		AVL_SETCHILD(node, which_child);
	}
	if (parent == NULL) {
		tree->avl_root = node;
		return;
	}
	parent->avl_child[which_child] = node;


	/*
	 * Since the subtree is now shorter, begin adjusting parent balances
	 * and performing any needed rotations.
	 */
	do {

		/*
		 * Move up the tree and adjust the balance
		 *
		 * Capture the parent and which_child values for the next
		 * iteration before any rotations occur.
		 */
		node = parent;
		old_balance = AVL_XBALANCE(node);
		new_balance = old_balance - avl_child2balance[which_child];
		parent = AVL_XPARENT(node);
		which_child = AVL_XCHILD(node);

		/*
		 * If a node was in perfect balance but isn't anymore then
		 * we can stop, since the height didn't change above this point
		 * due to a deletion.
		 */
		if (old_balance == 0) {
			AVL_SETBALANCE(node, new_balance);
			break;
		}

		/*
		 * If the new balance is zero, we don't need to rotate
		 * else
		 * need a rotation to fix the balance.
		 * If the rotation doesn't change the height
		 * of the sub-tree we have finished adjusting.
		 */
		if (new_balance == 0)
			AVL_SETBALANCE(node, new_balance);
		else if (!avl_rotation(tree, node, new_balance))
			break;
	} while (parent != NULL);
}

/*
 * initialize a new AVL tree
 */
void
avl_create(avl_tree_t *tree, int (*compar) (const void *, const void *),
    size_t size, size_t offset)
{
	ASSERT(tree);
	ASSERT(compar);
	ASSERT(size > 0);
	ASSERT(size >= offset + sizeof (avl_node_t));
#ifdef _LP64
	ASSERT((offset & 0x7) == 0);
#endif

	tree->avl_compar = compar;
	tree->avl_root = NULL;
	tree->avl_numnodes = 0;
	tree->avl_size = size;
	tree->avl_offset = offset;
}

/*
 * Delete a tree.
 */
/* ARGSUSED */
void
avl_destroy(avl_tree_t *tree)
{
	ASSERT(tree);
	ASSERT(tree->avl_numnodes == 0);
	ASSERT(tree->avl_root == NULL);
}


/*
 * Return the number of nodes in an AVL tree.
 */
ulong_t
avl_numnodes(avl_tree_t *tree)
{
	ASSERT(tree);
	return (tree->avl_numnodes);
}


#define	CHILDBIT	(1L)

/*
 * Post-order tree walk used to visit all tree nodes and destroy the tree
 * in post order. This is used for destroying a tree w/o paying any cost
 * for rebalancing it.
 *
 * example:
 *
 *	void *cookie = NULL;
 *	my_data_t *node;
 *
 *	while ((node = avl_destroy_nodes(tree, &cookie)) != NULL)
 *		free(node);
 *	avl_destroy(tree);
 *
 * The cookie is really an avl_node_t to the current node's parent and
 * an indication of which child you looked at last.
 *
 * On input, a cookie value of CHILDBIT indicates the tree is done.
 */
void *
avl_destroy_nodes(avl_tree_t *tree, void **cookie)
{
	avl_node_t	*node;
	avl_node_t	*parent;
	int		child;
	void		*first;
	size_t		off = tree->avl_offset;

	/*
	 * Initial calls go to the first node or it's right descendant.
	 */
	if (*cookie == NULL) {
		first = avl_first(tree);

		/*
		 * deal with an empty tree
		 */
		if (first == NULL) {
			*cookie = (void *)CHILDBIT;
			return (NULL);
		}

		node = AVL_DATA2NODE(first, off);
		parent = AVL_XPARENT(node);
		goto check_right_side;
	}

	/*
	 * If there is no parent to return to we are done.
	 */
	parent = (avl_node_t *)((uintptr_t)(*cookie) & ~CHILDBIT);
	if (parent == NULL) {
		if (tree->avl_root != NULL) {
			ASSERT(tree->avl_numnodes == 1);
			tree->avl_root = NULL;
			tree->avl_numnodes = 0;
		}
		return (NULL);
	}

	/*
	 * Remove the child pointer we just visited from the parent and tree.
	 */
	child = (uintptr_t)(*cookie) & CHILDBIT;
	parent->avl_child[child] = NULL;
	ASSERT(tree->avl_numnodes > 1);
	--tree->avl_numnodes;

	/*
	 * If we just did a right child or there isn't one, go up to parent.
	 */
	if (child == 1 || parent->avl_child[1] == NULL) {
		node = parent;
		parent = AVL_XPARENT(parent);
		goto done;
	}

	/*
	 * Do parent's right child, then leftmost descendent.
	 */
	node = parent->avl_child[1];
	while (node->avl_child[0] != NULL) {
		parent = node;
		node = node->avl_child[0];
	}

	/*
	 * If here, we moved to a left child. It may have one
	 * child on the right (when balance == +1).
	 */
check_right_side:
	if (node->avl_child[1] != NULL) {
		ASSERT(AVL_XBALANCE(node) == 1);
		parent = node;
		node = node->avl_child[1];
		ASSERT(node->avl_child[0] == NULL &&
		    node->avl_child[1] == NULL);
	} else {
		ASSERT(AVL_XBALANCE(node) <= 0);
	}

done:
	if (parent == NULL) {
		*cookie = (void *)CHILDBIT;
		ASSERT(node == tree->avl_root);
	} else {
		*cookie = (void *)((uintptr_t)parent | AVL_XCHILD(node));
	}

	return (AVL_NODE2DATA(node, off));
}