1、AVL-Trees,COMP171 Fall 2005,Balanced binary tree,The disadvantage of a binary search tree is that its height can be as large as N-1 This means that the time needed to perform insertion and deletion and many other operations can be O(N) in the worst case We want a tree with small height A binary tree
2、 with N node has height at least (log N) Thus, our goal is to keep the height of a binary search tree O(log N) Such trees are called balanced binary search trees. Examples are AVL tree, red-black tree.,AVL tree,Height of a node The height of a leaf is 1. The height of a null pointer is zero. The hei
3、ght of an internal node is the maximum height of its children plus 1Note that this definition of height is different from the one we defined previously (we defined the height of a leaf as zero previously).,AVL tree,An AVL tree is a binary search tree in which for every node in the tree, the height o
4、f the left and right subtrees differ by at most 1.,AVL property violated here,AVL tree,Let x be the root of an AVL tree of height h Let Nh denote the minimum number of nodes in an AVL tree of height h Clearly, Ni Ni-1 by definition We haveBy repeated substitution, we obtain the general formThe bound
5、ary conditions are: N1=1 and N2 =2. This implies that h = O(log Nh). Thus, many operations (searching, insertion, deletion) on an AVL tree will take O(log N) time.,Rotations,When the tree structure changes (e.g., insertion or deletion), we need to transform the tree to restore the AVL tree property.
6、 This is done using single rotations or double rotations.,x,y,A,B,C,Before Rotation,After Rotation,e.g. Single Rotation,Rotations,Since an insertion/deletion involves adding/deleting a single node, this can only increase/decrease the height of some subtree by 1 Thus, if the AVL tree property is viol
7、ated at a node x, it means that the heights of left(x) ad right(x) differ by exactly 2. Rotations will be applied to x to restore the AVL tree property.,Insertion,First, insert the new key as a new leaf just as in ordinary binary search tree Then trace the path from the new leaf towards the root. Fo
8、r each node x encountered, check if heights of left(x) and right(x) differ by at most 1. If yes, proceed to parent(x). If not, restructure by doing either a single rotation or a double rotation next slide. For insertion, once we perform a rotation at a node x, we wont need to perform any rotation at
9、 any ancestor of x.,Insertion,Let x be the node at which left(x) and right(x) differ by more than 1 Assume that the height of x is h+3 There are 4 cases Height of left(x) is h+2 (i.e. height of right(x) is h) Height of left(left(x) is h+1 single rotate with left child Height of right(left(x) is h+1
10、double rotate with left child Height of right(x) is h+2 (i.e. height of left(x) is h) Height of right(right(x) is h+1 single rotate with right child Height of left(right(x) is h+1 double rotate with right child,Note: Our test conditions for the 4 cases are different from the code shown in the textbo
11、ok. These conditions allow a uniform treatment between insertion and deletion.,Single rotation,The new key is inserted in the subtree A. The AVL-property is violated at xheight of left(x) is h+2height of right(x) is h.,Single rotation,Single rotation takes O(1) time. Insertion takes O(log N) time.,T
12、he new key is inserted in the subtree C. The AVL-property is violated at x.,5,Insert 0.8,AVL Tree,8,0.8,x,y,A,B,C,After rotation,Double rotation,The new key is inserted in the subtree B1 or B2. The AVL-property is violated at x. x-y-z forms a zig-zag shape,also called left-right rotate,Double rotati
13、on,The new key is inserted in the subtree B1 or B2. The AVL-property is violated at x.,also called right-left rotate,5,Insert 3.5,AVL Tree,8,1,After Rotation,An Extended Example,Insert 3,2,1,4,5,6,7, 16,15,14,Single rotation,Single rotation,Single rotation,Single rotation,Double rotation,Double rota
14、tion,Deletion,Delete a node x as in ordinary binary search tree. Note that the last node deleted is a leaf. Then trace the path from the new leaf towards the root. For each node x encountered, check if heights of left(x) and right(x) differ by at most 1. If yes, proceed to parent(x). If not, perform
15、 an appropriate rotation at x. There are 4 cases as in the case of insertion. For deletion, after we perform a rotation at x, we may have to perform a rotation at some ancestor of x. Thus, we must continue to trace the path until we reach the root.,Deletion,On closer examination: the single rotation
16、s for deletion can be divided into 4 cases (instead of 2 cases) Two cases for rotate with left child Two cases for rotate with right child,Single rotations in deletion,rotate with left child,In both figures, a node is deleted in subtree C, causing the height to drop to h. The height of y is h+2. Whe
17、n the height of subtree A is h+1, the height of B can be h or h+1. Fortunately, the same single rotation can correct both cases.,Single rotations in deletion,rotate with right child,In both figures, a node is deleted in subtree A, causing the height to drop to h. The height of y is h+2. When the hei
18、ght of subtree C is h+1, the height of B can be h or h+1. A single rotation can correct both cases.,Rotations in deletion,There are 4 cases for single rotations, but we do not need to distinguish among them. There are exactly two cases for double rotations (as in the case of insertion) Therefore, we can reuse exactly the same procedure for insertion to determine which rotation to perform,
