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O(log 2 n)

The average height of a BST :. O(log 2 n). f(n) : The average internal path length of an n-node BST. f(3) : (3+2+3) / 3 = 2.67. 0. 0. 1. 1. 1. 1. 2. 2. 0+1+1= 2. 0+1+2= 3. 0+1+2= 3.

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O(log 2 n)

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  1. The average height of a BST : O(log2n) f(n) : The average internal path length of an n-node BST f(3) : (3+2+3) / 3 = 2.67 0 0 1 1 1 1 2 2 0+1+1=2 0+1+2=3 0+1+2=3 IT 279

  2. Adel’son-Vels’kii and Landis, Soviet Mathematic Doklady, 3:1259-1263, 1962 C. Crane, D. Knuth, et al in 1970’s AVL Tree A BST in which the height difference between the two children of any node is always less than 2. +1 +1 -1 j k j+1 k+1 h h+1 IT 279

  3. AVL Tree -- an example of Dynamic Tree We dynamicallymaintain the properties of AVL-tree when we insert (remove) a node by four different operations (rotations) Dynamic Tree Static Tree Huffman Code -- an example of Static Tree We staticallyanalyze the code and build up an optimal tree for retrieving the code words. IT 279

  4. performed at the bad node where the difference between the heights of its two children is bigger than 1. Four operations : (rotations) If a node is bad caused by: then perform • Right-child’s Right-child • Left-child’s Left-child • Right-child’s Left-child • Left-Child’s Right-child • RR rotation • LL rotation • RL rotation • LR rotation IT 279

  5. No Rotation is Needed +1 0 -1 0 IT 279

  6. RR Rotations: +2 +1 IT 279

  7. RR Rotation: +2 R +0 R +0 +1 IT 279

  8. LL Rotation: L -2 +0 L -1 +0 IT 279

  9. Rotations: RL +1 +2 0 h+1 +1 -1 0 0 h+1 +0 -1 h h IT 279

  10. Rotations: RL RL Rotation: R 0 +2 L h+1 -1 0 +1 h+1 -1 h h h+1 h+1 h h IT 279

  11. Rotations: LR LR Rotation: L -2 0 1 R h+1 0 +1 h+1 -1 -1 h h h+1 h+1 h h IT 279

  12. Rotate this sub-tree first +2 R Could be RR or RL, depending on what happens in the blue sub-tree. R +2 Afterwards, examine the red node again to see is another rotation is needed. IT 279

  13. Rotations: LL +2 +1 +1 h+1 -1 -2 -1 h -2 -1 0 h-1 h -1 0 h-1 h-1 IT 279

  14. Example: 10 20 30 10 RR Rotation: 20 20 10 30 30 IT 279

  15. Example: 40 50 20 RR Rotation: 20 10 30 10 40 40 50 30 50 IT 279

  16. Example: 35 20 R 30 RL Rotation: 10 40 40 L 20 30 50 10 50 35 35 IT 279

  17. Possible complications Re-assign the links +2 h+1 -1 h+1 -1 h h Tracking the heights and balance-factors IT 279

  18. RR rotation in C++ a=t +2 R b template<typename T> // RR rotation on t; TreeNode<T> * AVLTree<T>::RR(TreeNode<T> * t) { TreeNode<T> * a = t, * b = t->right; a->right = b->left; b->left = a; a->height -= 2; return b; } h R +1 h h height is an extra member variable in the TreeNode. IT 279

  19. RL rotation in C++ a=t template<typename T> // RL rotation on t; TreeNode<T> * AVLTree<T>::RL(TreeNode<T> * t) { TreeNode<T> *a, *b, *c; a = t; b = t->right; c = t->right->left; a->right = c->left; b->left = c->right; c->left = a; c->right = b; c->height++; b->height--; a->height-=2; return c; } b R +2 L c h+1 -1 h+1 -1 h h IT 279

  20. AVL h: Average Heights n Random Keys IT 279

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