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Tree Balancing: AVL Trees

Tree Balancing: AVL Trees. Dr. Yingwu Zhu. Recall in BST. The insertion order of items determine the shape of BST Balanced: search T(n)=O(logN) Unbalanced: T(n) = O(n) Key issue: A need to keep a BST balanced! Introduce AVL trees, by Russian mathematican. AVL Tree Definition.

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Tree Balancing: AVL Trees

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  1. Tree Balancing: AVL Trees Dr. Yingwu Zhu

  2. Recall in BST • The insertion order of items determine the shape of BST • Balanced: search T(n)=O(logN) • Unbalanced: T(n) = O(n) • Key issue: • A need to keep a BST balanced! • Introduce AVL trees, by Russian mathematican

  3. AVL Tree Definition • First, a BST • Second, height-balance property: balance factor of each node is 0, 1, or -1 • Question: what is balance factor? BF = Height of the left subtree – height of the right subtree Height: # of levels in a subtree/tree

  4. Determine balance factor A B D C E G F H

  5. ADT: AVL Trees • Data structure to implement Balance factor Data Left Right

  6. ADT: AVL Trees • Basic operations • Constructor, search, travesal, empty • Insert: keep balanced! • Delete: keep balanced! • See P842 for class template • Similar to BST

  7. Example RI Insert “DE”, what happens? Need rebalancing? PA

  8. Basic Rebalancing Rotation • Single rotation: • Right rotation: the inserted item is on the left subtree of left child of the nearest ancestor with BF of 2 • Left rotation: the inserted item is on the right subtree of right child of the nearest ancestor with BF of -2 • Double rotation • Left-right rotation: the inserted item is on the right subtree of left child of the nearest ancestor with BF of 2 • Right-left rotation: the inserted item is on the left subtree of right child of the nearest ancestor with BF of -2

  9. How to perform rotations • Rotations are carried out by resetting links • Two steps: • Determine which rotation • Perform the rotation

  10. Right Rotation • Key: identify the nearest ancestor of inserted item with BF +2 • A: the nearest ancestor. B: left child • Step1: reset the link from parent of A to B • Step2: set the left link of A equal to the right link of B • Step3: set the right link of B to A • Examples

  11. new A B L(A) L(B) R(B) h

  12. Exercise #1 10 4 How about insert 8 20 6 9

  13. Left Rotation • Step1: reset the link from parent of A to B • Step2: set the right link of A to the left link of B • Step3: set the left link of B to A • Examples

  14. new Example A B L(A) h L(B) R(B)

  15. Exercise #2 7 18 What if insert 5 15 10 20

  16. Exercise #3 12 1 How about insert 8 16 4 10 14 2 6

  17. Basic Rebalancing Rotation • Double rotation • Left-right rotation: the inserted item is on the right subtree of left child of the nearest ancestor with BF of 2 • Right-left rotation: the inserted item is on the left subtree of right child of the nearest ancestor with BF of -2

  18. Double Rotations • Left-right rotation • Right-left rotation • How to perform? 1. Rotate child and grandchild nodes of the ancestor 2. Rotate the ancestor and its new child node

  19. Example A C What if insert B

  20. new Example A B R(A) h+1 C L(B) h+1 L(C) R(C) h

  21. Exercise #4 12 7 How about insert 8 16 4 10 14 2 6

  22. Exercise #5 4 2 6 15 What if insert 1 3 5 7 16

  23. Exercise #5 4 2 6 14 What if insert 1 3 5 15 7 16

  24. Summary on rebalancing • The key is you need to identify the nearest ancestor of inserted item • Determine the rotation by definition • Perform the rotation

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