1. Tree Rotations & Splay Trees

2. BST Structure BST's only perform well when balanced
But common cases lead to unbalanced trees

3. Restoring Balance
How could this unbalanced tree be balanced?

4. Restoring Balance
How could this unbalanced tree be balanced?

5. Rotation Parent node become child, child becomes parent
Right rotation : Everyone moves right (clockwise)
Left rotation : Counter clockwise

6. Rotation Description Describe rotations by parent Rotate 38 right

7. Child Issues Want to rotate 38 right (make 21 parent)
Where does 26 go?

8. Child Issues
Right child of old child  left child of old parent

9. Right Rotation

10. Left Rotation

11. Multi Step
Want to balance this tree:

12. Multi Step Want to balance this tree:
Rotate root right is just as bad:

13. Multi Step
Want to balance zig-zag pattern

14. Multi Step
Want to balance zig-zag pattern
Rotate child of root left

15. Multi Step Want to balance zig-zag pattern Rotate child of root left
Rotate root right

16. Deciding What to Rotate
Figuring out where to rotate is tough
Need more info than available locally in BST
AVL, RB

17. Deciding What to Rotate
Figuring out where to rotate is tough
Need more info than available locally in BST
AVL, RB
Alternative:
Keep recent nodes at top of tree
Do a pretty good job balancing

18. Recent to Top
Simple Rotating upwards to root does not balance tree:

19. Splay Tree Splaying : move node to root using rotates
Splay Tree : after any access, move node to root
O(1) to access same element again
Faster to access nearby values
Amortized O(logn) average access

20. How it works:
Insert 1-6 in order BST: /BST.html
Insert 1-6 in order Splay Tree: /SplayTree.html

21. Zig-Zig and Zig-Zag Rotations done in 3 ways
Node is right below root: Zig (Rotate left or right)

22. Zig-Zig and Zig-Zag Rotations done in 3 ways
Path to node is left-left or right-right: Zig-Zig (Rotate grandparent, then rotate parent)

23. Zig-Zig and Zig-Zag Rotations done in 3 ways
Path to node is left-right or right-left: Zig-Zag (Rotate parent one way, then grand parent other way)

24. Sample:
Splay 42

25. Sample:
Splay 42
It is left-left child : Zig-Zig

26. Sample:
Splay 42
It is left-left child : Zig-Zig
Rotate 52 right

27. Sample: Splay 42 It is left-left child : Zig-Zig
Rotate 52 right, rotate 47 right

28. Sample:
Splay 42
Now it is a left-right child : Zig-Zag

29. Sample: Splay 42 Now it is a left-right child : Zig-Zag
35 Rotates left

30. Sample: Splay 42 Now it is a left-right child : Zig-Zag
35 Rotates left, 70 rotate right