Avltrees

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Avltrees

  1. 1. AVL Trees 1
  2. 2. Presentation Data Structure & Algorithms Presented By: Group 11 By Komal 6622 Tayyaba 6648 Maryam 6626 Kanwal 6619 Mahnoor 6623 AVL Trees 2
  3. 3. AVL Trees 6 v 8 3 z 4 AVL Trees 3
  4. 4. Contents 1) Defination 2) Why we need? 3) Difference beteen AVL and Non AVL by example 4) Techniques with examples 5) Advantage and Disadvantage 6) Time complexity AVL Trees 4
  5. 5.   AVL Trees: AVL trees were invented in 1962 by two Russian scientist G.M Adelson Velsky & E.M Landis. Definition: An AVL tree is a binary search tree in which the balance factor of every node is either 0 or +1 or -1 .
  6. 6. Height Of A Tree: The height of a binary tree is the maximum path length from root to the leaf Level 0 Level 1 Level 2 Level 3 The height of this tree is= 3 AVL Trees 6
  7. 7. Balance Factor:  It is defined as the difference b/w the heights of the node’s left & right sub trees  Balance factor=ht of left sub tree – ht of right sub tree.  ht=height AVL Trees 7
  8. 8. Why we need AVL Tree? o To keep binary search tree perfectly balanced we need AVL Tree with O(log N) complexity… AVL Trees 8
  9. 9. Example : AVL tree BST ( not AVL tree ) 1 2 0 0 0 -1 0 0 0 0 0 0 -1 0 0
  10. 10. Techniques to make AVL Tree: If an insertion of a new node makes an AVL tree unbalanced , we transform the tree by a rotation there are four types of rotation we have: 1. 2. 3. 4. Single right rotation or R-rotation Single left rotation or L-rotation Double right-left rotation or RL-rotation Double left-right rotation or LR-rotation AVL Trees 10
  11. 11.   1.Straight line with positive unbalanced. apply Right rotation for unbalanced node. +2 +1 0 2 4 5 0 4 0 2 0 5
  12. 12.   2.Straight line with negative unbalanced. apply left rotation for unbalanced node. 5 -2 0 6 -1 7 0 5 0 6 0 7
  13. 13.   3.Curved line with positive unbalanced. apply left-right rotation. Right rotation for unbalanced node and left rotation for the nearest node. For example: C,A,B +2 C C 0 B B A -1 B 0 A 0 A C 0
  14. 14.   4.Curved line with negative unbalanced. apply right-left rotation. Left rotation for unbalanced node and right rotation for the nearest node. For example: A,C,B A -2 C B 0 A 0 B B +1 C 0 A C 0
  15. 15. Advantages &Disadvantages  The advantage of an AVL tree is that it is always balanced, guaranteeing the O(logn) speed of the Binary Search algorithm.  The disadvantages the complex rotations used by the insertion and removal AVL Trees 15
  16. 16. Running Times for AVL Trees find is O(log n)  height of tree is O(log n) insert is O(log n)  initial find is O(log n) remove is O(log n)  initial remove is O(log n) AVL Trees 16
  17. 17. Thank You  AVL Trees 17

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