AVL Trees. (AVL Trees) Data Structures and Programming Spring / 17
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1 AVL Trees (AVL Trees) Data Structures and Programming Spring / 17
2 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 with N node has height at least Θ(log N) Thus, our goal is to keep the height of a binary search tree O(logN) Such trees are called balanced binary search trees. Examples are AVL tree, red-black tree. (AVL Trees) Data Structures and Programming Spring / 17
3 AVL Tree (Georgy Adelson-Velsky and Evgenii Landis tree, 1962) Height of a node The height of a leaf is 1. The height of a null pointer is zero. The height of an internal node is the maximum height of its children plus 1 Note that this definition of height is different from the one we defined previously (we defined the height of a leaf as zero previously). (AVL Trees) Data Structures and Programming Spring / 17
4 AVL Tree (Cont d) An AVL tree is a binary search tree in which for every node in the tree, the height of the left and right subtrees differ by at most 1. (AVL Trees) Data Structures and Programming Spring / 17
5 AVL Tree (Cont d) Let x be the root of an AVL tree of height h Let N h denote the minimum number of nodes in an AVL tree of height h Clearly, N i N i 1 by definition We have N h N h 1 + N h N h > 2N h 2 By repeated substitution, we obtain the general form N h > 2 i N h 2i The boundary conditions are: N 1 = 1 and N 2 = 2. This implies that h = O(logN h ). More precisely, the height of an n-node AVL tree is approx log 2 n. Thus, many operations (searching, insertion, deletion) on an AVL tree will take O(logN) time. (AVL Trees) Data Structures and Programming Spring / 17
6 Smallest AVL Tree (AVL Trees) Data Structures and Programming Spring / 17
7 Rotations When the tree structure changes (e.g., insertion or deletion), we need to transform the tree to restore the AVL tree property. This is done using single rotations or double 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 violated at a node x, it means that the heights of left(x) and right(x) differ by exactly 2. Rotations will be applied to x to restore the AVL tree property. (AVL Trees) Data Structures and Programming Spring / 17
8 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. 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, 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 won t need to perform any rotation at any ancestor of x. (AVL Trees) Data Structures and Programming Spring / 17
9 Single rotation The AVL-property is violated at x height of left(x) is h+2 height of right(x) is h. (AVL Trees) Data Structures and Programming Spring / 17
10 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 (AVL Trees) Data Structures and Programming Spring / 17
11 Example - Rebalance after Insertion (AVL Trees) Data Structures and Programming Spring / 17
12 Another Example (AVL Trees) Data Structures and Programming Spring / 17
13 Another Example (Cont d) (AVL Trees) Data Structures and Programming Spring / 17
14 Deletion The single rotations for deletion can be divided into 4 cases In the following, a node is deleted in subtree C, causing the height to drop to h. The height of y is h+2. When 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. Imbalance may propagate upward so that many rotations may be needed. (AVL Trees) Data Structures and Programming Spring / 17
15 Example - Rebalance after Deletion (Single Rotation) (AVL Trees) Data Structures and Programming Spring / 17
16 Example - Rebalance after Deletion (Double Rotation) 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 (AVL Trees) Data Structures and Programming Spring / 17
17 Pros and Cons of AVL Trees Pros and Cons of AVL Trees 1 Search is O(log N) since AVL trees are always balanced. 2 Insertion and deletions are also O(log n) 3 The height balancing adds no more than a constant factor to the speed of insertion. Arguments against using AVL trees: 1 Difficult to program and debug; more space for balance factor. 2 Asymptotically faster but rebalancing costs time. 3 Most large searches are done in database systems on disk and use other structures (e.g. B-trees). 4 May be OK to have O(N) for a single operation if total run time for many consecutive operations is fast (e.g. Splay trees). (AVL Trees) Data Structures and Programming Spring / 17
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