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AVL TREES

AVL TREES. By Asami Enomoto CS 146. AVL Tree is…. named after A delson- V elskii and L andis the first dynamically balanced trees to be propose Binary search tree with balance condition in which the sub-trees of each node can differ by at most 1 in their height.

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AVL TREES

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  1. AVL TREES By Asami Enomoto CS 146

  2. AVL Tree is… • named after Adelson-Velskii and Landis • the first dynamically balanced trees to be propose • Binary search tree with balance condition in which the sub-trees of each node can differ by at most 1 in their height

  3. Definition of a balanced tree • Ensure the depth = O(log N) • Take O(log N) time for searching, insertion, and deletion • Every node must have left & right sub-trees of the same height

  4. An AVL tree has the following properties: Sub-trees of each node can differ by at most 1 in their height Every sub-trees is an AVL tree

  5. AVL tree? YES Each left sub-tree has height 1 greater than each right sub-tree NO Left sub-tree has height 3, but right sub-tree has height 1

  6. Insertion and Deletions • It is performed as in binary search trees • If the balance is destroyed, rotation(s) is performed to correct balance • For insertions, one rotation is sufficient • For deletions, O(log n) rotations at most are needed

  7. Single Rotation move ① up a level and ② down a level left sub-tree is two level deeper than the right sub-tree

  8. Double Rotation Move ② up two levels and ③ down a level Left sub-tree is two level deeper than the right sub-tree

  9. Insertion Insert 6 Imbalance at 8 Perform rotation with 7

  10. Deletion Delete 4 Imbalance at 3 Perform rotation with 2 Imbalance at 5 Perform rotation with 8

  11. Key Points • AVL tree remain balanced by applying rotations, therefore it guarantees O(log N) search time in a dynamic environment • Tree can be re-balanced in at most O(log N) time

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