1. AVL Trees

2. Knowing More How many nodes? – Determine on demand

3. Knowing More How many nodes? – Determine on demand int size(Node curNode) if curNode is null, return 0 else return 1 + size(leftChild) + size(rightChild)

4. Knowing More How many nodes? – Determine on demand – T(n) = 2T(n/2) + 1 O(n) int size(Node curNode) if curNode is null, return 0 else return 1 + size(leftChild) + size(rightChild)

5. Knowing More How many nodes? – Store in tree Update on insert / delete O(1)

6. Knowing More How balanced are we?

7. Knowing More How balanced are we? – How many nodes off each side? 3 7

8. Knowing More How balanced are we? – How many nodes off each side????? 7 7

9. Knowing More How balanced are we? – What is the longest path in each direction? 3 3

10. Knowing More How balanced are we? – What is the longest path in each direction? Tree ops dependent on longest path – Consistent depth is key 3 3

11. Knowing More Height of node – Determine on demand int depth(Node curNode) if curNode is null, return -1 else return 1 + max( depth(leftChild), depth(rightChild))

12. Knowing More Height of node – Determine on demand – T(n) = 2T(n/2) + 1 O(n) per node int depth(Node curNode) if curNode is null, return -1 else return 1 + max( depth(leftChild), depth(rightChild))

13. Knowing More Height of node – Store in each node Update on insert / delete O(1) 0 0 0 1 1 2 3

14. AVL Trees Georgii Adelson-Velsky Evgenii Mikhailovich Landis

15. AVL Rules Every node has balance factor Balance = Height(left child) – Height(right child) Height of Null = -1 0 0 0 1 2 3

16. AVL Rules Node must have balance factor of -1, 0 or 1 – Rotate to fix bad nodes

17. AVL Rules Worst case: Height01234 Nodes124712

18. AVL Rules Worst case: Height01234 Nodes124712

19. AVL Rules Worst case: Height01234 Nodes124712

20. AVL Rules Worst case: Height01234 Nodes124712

21. AVL Rules Worst case: Nodes at height h N h = N h-1 + N h-2 + 1 Height01234 Nodes124712

22. AVL Rules Worst case:

23. So…. Guaranteed log(n) height – O(logn) Insert/Delete/Remove – But… Higher constant factors for insert/remove

24. Maintaining Balance Inserting/deleting node changes balance by 1 – Affects whole chain

25. Maintaining Balance -2 or +2 needs to be balanced 2 cases – Outside Cases (single rotation) : Longest chain is left-left or right-right of unbalanced – Inside Cases (double rotation) : Longest chain is left-right or right-left of unbalanced

26. Case 1 A valid AVL subtree f AD G h h h b h h+1

27. Case 1 Insertion creates problem on left-left or right- right f A D G h h+1 h b h+2

28. Rotate away from problem Case 1 f A D G h h+1 h b

29. Case 2 Problem created on left-right or right-left path f A G h h h b C E d h - 1 h+1

30. Case 2 Problem created on left-right or right-left path f A G h h h+1 b C E d h h - 1 h+2

31. Case 2 Do Zig-Zag Rotation f A G h h h+2 b C E d h h - 1 h+1

32. Case 2 Do Zig-Zag Rotation f A G h h h+2 b C E d h h - 1 h+1

33. Deletions Deletions like rotations – rotate to restore balance

34. Deletions Deletions like rotations – rotate to restore balance – May unbalance multiple nodes

35. Deletions Deletions like rotations – rotate to restore balance – May unbalance multiple nodes

36. Deletions Deletions like rotations – rotate to restore balance – May unbalance multiple nodes