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In this lecture we use AVL trees which is a simple and ef cient data structure to maintain balance and is also the rst that has been proposed It is named after its inventors G M Adelson Velskii and E M Landis who described it in 1962 LECTURENOTESMARCH22 2011 AVL Trees L18 2 2 The Height Invariant
span class result type, 4 1 0 6 3 8 2 0 7 10 11 0 12 1 0 An AVL tree The shallowness bound Let S h the minimum number of nodes in an AVL tree of height h If we can prove that S h grows exponentially in h then a tree with n nodes has a logarithmic height Step 1 Define S h inductively using AVL property
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CSE 373 Data Structures and Algorithms Warm Up An AVL tree is a binary search tree that also meets the following rule AVL condition For every node the height of its left subtree and right subtree differ by at most 1 pollEV cse373su19 Is this a valid AVL tree Is this a valid AVL tree 3 2 6 5 4 7
span class result type, The AVL Tree Data Structure An AVL tree is a self balancing binary search tree Structural properties Binary tree property same as BST Order property same as for BST Balance property balance of every node is between 1 and 1 Need to keep track of height of every node and maintain balance as we perform operations AVL Trees Insert
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span class result type, In this lecture we use AVL trees which is a simple and efficient data structure to maintain balance and is also the first that has been proposed It is named after its inventors G M Adelson Velskii and E M Landis who described it in 1962 Additional Resources
AVL Tree
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span class result type A binary search tree is a binary tree T such that each internal node stores an item k e of a dictionary keys stored at nodes in the left subtree of v are less than or equal to k Keys stored at nodes in the right subtree of v are greater than or equal to k External nodes do not hold elements but serve as place holders
Pin By Yunrui Zhang On Data Structures And Algorithms Binary Tree
The AVL Tree Data Structure 4 2 6 10 12 5 11 8 7 9 13 14 Structural properties 1 Binary tree property 2 Balance property balance of every node is between 1 and 1 Result Worst case depth is O logn Ordering property Same as for BST 15 Spring 2010 CSE332 Data Abstractions Spring 2010 CSE332 Data Abstractions 3 AVL Tree Deletion span class result type. An empty tree has height 1 A tree with a single node has height 0 A binary search tree is said to be AVL balanced if The difference in the heights between the left and right sub trees is at most 1 and Both sub trees are themselves AVL trees AVL trees with 1 2 3 and 4 nodes Here is a larger AVL tree 42 nodes The AVL Tree Data Structure 4 2 6 10 12 5 11 8 7 9 13 14 Structural properties 1 Binary tree property 0 1 or 2 children 2 Heights of left and right Proof means that if we have an AVL tree then find is O log n But as we insert and delete elements we need to 1 Track balance 2 Detect imbalance 3 Restore balance 1 19 2011 13 2 9 5 10 7
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