Introduction to Algorithms (AVL Trees, AVL Sort)

Recall: Binary Search Trees

• rooted binary tree
• each node has:
  • key
  • left pointer
  • right pointer
  • parent pointer
• BST property
• the height of node = length of the longest downward path to a leaf

The importance of being balanced

• BSTs support insert, delete, min, max, next-larger, next-smaller, etc. in O(h) time, where h = height of the tree (= height of root).
• keep BST small <=> keep the height of BST small <=> keep BST balanced 

AVL Trees

requires heights of left & right children of every node to differ by at most ±1

• treat nil tree as height - 1
• each node stores its height (DATA STRUCTURE AUGMENTATION) (like subtree size) (alternatively, can just store the difference in heights)

Balance

Worst when every node differs by 1

let  = (min.) # nodes in height-h AVL tree

 AVL insert

1. simple BST insert
2. fix AVL property ​​​​

   

    

   1. suppose x is the lowest node violating AVL
   2. suppose x is right-heavy
   3. if x's right child is right heavy or balanced, LR(x)
   4. else RR(x), LR(x)