Foundation of Computing Systems

1. Foundation of Computing Systems Lecture 8 Trees: Part V

2. Height Balanced Binary Trees • Average search time in a binary search tree • A binary search tree should be with a minimum value of average search time ti = Number of comparisons for the i-th element n = Total number of elements in the binary search tree IT 60101: Lecture #8

3. Some Sample Binary Search Trees G = 6.50   IT 60101: Lecture #8

4. Some Sample Binary Search Trees G = 4.00   G = 3.50   IT 60101: Lecture #8

5. Some Sample Binary Search Trees G = 3.08   G = 3.08   IT 60101: Lecture #8

6. Some Sample Binary Search Trees G = 3.16   IT 60101: Lecture #8

7. Height Balanced Binary Trees • How to find a binary search tree with a minimum value of average search time ? • Solution • Height balanced binary search tree • Also called AVL tree • Concept of balance factor of a node bf = Height of the left sub-tree (hL) – Height of the right sub-tree (hR) IT 60101: Lecture #8

8. Height Balanced Binary Trees • Definition A binary search tree is said to be height balanced binary search tree if all its nodes have a balance factor of 1, 0 or –1. That is, for all nodes |bf| = |hL – hR| ≤ 1 IT 60101: Lecture #8

9. Height Balanced Binary Trees: Example IT 60101: Lecture #8

10. Managing Unbalanced BST • Steps • Insert node into a binary search tree • Compute the balance factors • Decide the pivot node • Balance the unbalance tree IT 60101: Lecture #8

11. Unbalanced to Balanced BST • AVL Rotations • G.M. Adelson-Velskii and E.M. Lendis (1962) • Case 1: Left-to-Left rotation • pivotleft-sub treeleft child • Case 2: Right-to-Right rotation • pivotright-sub treeright child • Case 3: Left-to-Right rotation • pivotleft-sub treeright child • Case 4: Right-to-Left rotation • pivotright-sub treeleft child IT 60101: Lecture #8

12. Left-to-Left Rotation IT 60101: Lecture #8

13. Left-to-Left Rotation IT 60101: Lecture #8

14. Left-to-Left Rotation IT 60101: Lecture #8

15. Left-to-Left Rotation IT 60101: Lecture #8

16. Right-to-Right Rotation IT 60101: Lecture #8

17. Right-to-Right Rotation IT 60101: Lecture #8

18. Left-to-Right Rotation • Also called double rotation • Rotation 1 (Right-to-Left) • Left sub-tree (BL) of the right child (B) of the left child of the pivot node (P) becomes the right sub-tree of the left child (A). • Left child (A) of the pivot node (P) becomes the left child of B. • Rotation 2 (Left-to-Right) • Right sub-tree (BR) of the right child (B) of the left child (A) of the PIVOT node (P) becomes the left sub-tree of P. • P becomes the right child of B. IT 60101: Lecture #8

19. Left-to-Right Rotation IT 60101: Lecture #8

20. Left-to-Right Rotation IT 60101: Lecture #8

21. Left-to-Right Rotation IT 60101: Lecture #8

22. Right-to-Left Rotation • It is also another double rotation • Rotation 1(Left-to-Right) • Right sub-tree (BR) of the left child (B) of the right child (A) of the pivot node (P) becomes the left sub-tree of A. • Right child (A) of the pivot node (P) becomes the right child of B. • Rotation 2 (Right-to-Left) • Left sub-tree (BL) of the right child (B) of the right child (A) of the pivot rode (P) becomes the right sub-tree of P. • P becomes the left child of B. IT 60101: Lecture #8

23. Right-to-Left Rotation IT 60101: Lecture #8

24. Right-to-Left Rotation IT 60101: Lecture #8

25. Right-to-Left Rotation IT 60101: Lecture #8

26. Implementation • For tricky ways to manage the pointers see the book • Classic Data Structures • Chapter 7 • PHI, 2nd Edn., 17th Reprint IT 60101: Lecture #8

27. Property of a HBT • Maximum height possible in an AVL tree with n number of nodes is given by • For the proof of this property see the book • Classic Data Structures • Chapter 7 • PHI, 2nd Edn., 17th Reprint IT 60101: Lecture #8