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Balanced Binary Search Trees Drawback of a binary search tree There are two basic ways to avoid having this happen: 1. 2. Average time to build a BST with n keys and average time to find an element in a BST. The internal path length of a tree is AVL trees An AVL tree is Balance factors: -1 indicates 0 indicates +1 indicates Rebalancing is done by using rotations. http://www.cse.hut.fi/en/research/SVG/TRAKLA2/exercises.shtml web site for algorithm animations Insertions in an AVL Tree Insertions: 1. 2. 3. Example AVL tree Insertions--Example Insert 10, 15, 5, 12, 20 into tree Inserting 7 and then 6 requires a double rotation Generalizing: Deletions in AVL trees Splay Trees amortized running time Strategy—every time a node is accessed, it must be moved. A splay tree is an example of a self-adjusting tree. Splaying The starting node for a splay depends on the type of access. 1. find— 2. insert— 3. delete— Let S be the start node of the splay (1) If S is the root, STOP (2) If S has a parent P, but no grandparent, (3) S has both a parent P and a grandparent G.