1. CO4301 – Advanced Games Development Week 11 AVL Trees
Gareth Bellaby

2. AVL tree
Georgy Adelson-Velsky and Evgenii Landis, who published it in their 1962 paper "An algorithm for the organization of information"

3. Self-balancing tree Another balanced tree.
As with the others, foundation is a Binary Search Tree (BST)
A height balanced BST.
An example of a Self-balancing tree

4. AVL tree Any subtree of an AVL tree will itself be an AVL tree.
Height of the tree is O(logn).
Implemented by maintaining an extra attribute in each node which records the height of the node.
Rule to maintain the height-balance property.

5. AVL tree
The height of a node v in a tree is the length of the longest path from v to an external node.
Height-Balance Property: For every internal node v of T, the heights of the children of v differ by at most 1.
Note, not every node is within a distance of 1 compared with every other node, but the children of each internal node

6. Operations
Insertion and removal operations for AVL trees are like those for a BST tree (or Red-Black tree)
Like the Red-Black tree an additional step is needed in order to maintain the property of the tree, in this case the Height-Balance Property.
If an insertion operation results in the the Height-Balance Property being broken the balance of the tree is repaired by a “search-and-repair” strategy.

7. Comparison to Red-Black
O(log n) time for the main set of operations.
AVL tree is more strictly balanced than a Red-Black Tree.
Because of this characteristic they are faster for lookup.
However, they are therefore also slower in practice for insertion and deletion because of the need to remain strictly balanced.

8. Example
Insert 60, 40, 70, 20, 50, 80

9. Example
If 90 was inserted then the height-balance property will be broken and the tree will need to be re-balanced

10. Insert There are 4 cases: Insert into right subtree of a right node
Insert into left subtree of a left node
Insert into right subtree of a left node
Insert into left subtree of a right node
In every case, the tree is rebalanced by a rotation, or by two rotations

11. Insert right of right

12. Insert right of right
Single left rotation

13. Insert left of left

14. Insert left of left
Single right rotation

15. Insert right of left

16. Insert right of left Double rotation: left, followed by right
First single left rotation around central node

17. Insert right of left
Right rotation

18. Insert left of right

19. Insert left of right Double rotation: right, followed by left
First single right rotation around central node

20. Insert left of right
Left rotation

21. Deletion Delete the node.
Travel up from the node and find the first unbalanced node.
Let C be the first unbalanced node
Let B be the larger height child of C
Let A be the larger height child of B.

22. Deletion
Balanced tree

23. Deletion
Delete 20. Unbalanced.

24. Deletion
4 cases
Left-Left: B is left child of C and A is left child of B
Left-Right: B is left child of C and A is right child of B
Right-Right: B is right child of C and A is right child of B
Right-Left: B is right child of C and A is left child of B

25. Deletion Left-Left
Right rotation on C

26. Deletion Left-Right Left rotation on B Right rotation on C

27. Deletion Left-Right
Right rotation on C

28. Deletion Right-Right
Left rotation on C

29. Deletion Right-Left Right rotation on B Left rotation on C

30. Deletion Right-Left
Left rotation on C

31. AVL tree Nice visualisation:

32. References
Leonidas J. Guibas and Robert Sedgewick, "A Dichromatic Framework for Balanced Trees“
Frank M. Carrano, Timothy Henry, Data Abstraction & Problem Solving with C++: Walls and Mirrors Paperback
Goodrich, Tamassia & Mount, Data Structures and Algorithms

33. Next Week Computational geometry Cormen, Chapter 33
Goodrich & Tamassia, Chapter 22
Convex Hulls
Segment intersection
Closest pair of points