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Understanding AVL Trees: Balancing Operations and Path Lengths

This text delves into the concept of AVL trees, a type of self-balancing binary search tree (BST), highlighting the average height and internal path length calculations. It discusses the critical balancing operations (right, left, right-left, and left-right rotations) that ensure height balance during node insertion and deletion. C++ implementations of these rotations are provided for better comprehension. Through mathematical insights and practical examples, this content serves as a comprehensive guide for those studying tree data structures.

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Understanding AVL Trees: Balancing Operations and Path Lengths

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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 ITK 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 ITK 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. ITK 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 ITK 279

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

  6. RR Rotations: +2 +1 ITK 279

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

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

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

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

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

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

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

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

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