![How does the AVL tree work?
First, insert the new key as a new leaf just as in
ordinary binary search tree
Then trace the path from the new leaf towards the
root. For each node x encountered, check if heights
of left(x) and right(x) differ by at most 1.
If yes, proceed to parent(x). If not, restructure by
doing either a single rotation or a double rotation
[next slide].
For insertion, once we perform a rotation at a node x,
we won’t need to perform any rotation at any
ancestor of x.](https://image.slidesharecdn.com/avltrees-240323085526-c2ebd460/85/Design-data-Analysis-Avl-Trees-pptx-by-piyush-sir-12-320.jpg)
Design data Analysis Avl Trees.pptx by piyush sir
- 1. AVL TREES
- 2. AVL TREES (Adelson-Velskii and Landis 1962) AVL tree is a self-balancing Binary Search Tree (BST) where the difference between heights of left and right subtrees cannot be more than one for all nodes. Key idea: if insertion or deletion get the tree out of balance then fix it immediately All operations insert, delete,… can be done on an AVL tree with N nodes in O(log N) time (average and worst case!)
- 3. Example: An example of an AVL tree where the heights are shown next to the nodes: 88 44 17 78 32 50 48 62 1 3 0 0 1 2 0 0
- 4. Insertion in an AVL Tree Insertion is as in a binary search tree (always done by expanding an external node)
- 5. Insertion in an AVL Tree Insertion is as in a binary search tree (always done by expanding an external node) Example: 44 17 78 32 50 88 48 62
- 6. Insertion in an AVL Tree Insertion is as in a binary search tree (always done by expanding an external node) Example: Insert node 54 44 17 78 32 50 88 48 62
- 7. Insertion in an AVL Tree Insertion is as in a binary search tree (always done by expanding an external node) Example: 44 17 78 32 50 88 48 62 54 Insert node 54 44 17 78 32 50 88 48 62
- 8. Insertion in an AVL Tree Insertion is as in a binary search tree (always done by expanding an external node) Example: Insert node 54 2 44 17 78 32 50 88 48 62 54 44 17 78 32 50 88 48 62
- 9. Insertion in an AVL Tree Insertion is as in a binary search tree (always done by expanding an external node) Example: Insert node 54 2 1 44 17 78 32 50 88 48 62 54 44 17 78 32 50 88 48 62
- 10. Insertion in an AVL Tree Insertion is as in a binary search tree (always done by expanding an external node) Example: Insert node 54 2 1 0 44 17 78 32 50 88 48 62 54 44 17 78 32 50 88 48 62
- 11. Insertion in an AVL Tree Insertion is as in a binary search tree (always done by expanding an external node) Example: Insert node 54 2 1 0 44 17 78 32 50 88 48 62 54 44 17 78 32 50 88 48 62 Unbalanced!!
- 12. How does the AVL tree work? First, insert the new key as a new leaf just as in ordinary binary search tree Then trace the path from the new leaf towards the root. For each node x encountered, check if heights of left(x) and right(x) differ by at most 1. If yes, proceed to parent(x). If not, restructure by doing either a single rotation or a double rotation [next slide]. For insertion, once we perform a rotation at a node x, we won’t need to perform any rotation at any ancestor of x.
- 13. Rotations Two types of rotations Single rotations two nodes are “rotated” Double rotations three nodes are “rotated”
- 14. Single Rotation (Right) Rotate x with left child y (pay attention to the resulting sub-trees positions)
- 15. Single Rotation (Left) Rotate x with right child y (pay attention to the resulting sub-trees positions)
- 16. Single Rotation - Example Tree is an AVL tree by definition. h h+1
- 17. Example h h+2 Node 02 added Tree violates the AVL definition! Perform rotation.
- 18. Tree has this form. h h h+1 A B C x y Example
- 19. Example – After Rotation Tree has this form. A B C x y
- 21. Single Rotation Sometimes a single rotation fails to solve the problem k2 k1 X Y Z k1 X Y Z k2 h+2 h h h+2 In such cases, we need to use a double-rotation
- 22. Double Rotations
- 23. Double Rotations
- 24. Double Rotation - Example Tree is an AVL tree by definition. h h+1 Delete node 94
- 25. Example AVL tree is violated. h h+2
- 26. Tree has this form. B1 B2 C A x y z Example
- 27. A B1 B2 C x y z Tree has this form After Double Rotation
- 29. Insertion The time complexity to perform a rotation is O(1) The time complexity to find a node that violates the AVL property is dependent on the height of the tree, which is log(N)
- 30. Deletion Perform normal BST deletion Perform exactly the same checking as for insertion to restore the tree property
- 31. Summary AVL Trees Maintains a Balanced Tree Modifies the insertion and deletion routine Performs single or double rotations to restore structure Guarantees that the height of the tree is O(logn) The guarantee directly implies that functions find(), min(), and max() will be performed in O(logn)
- 32. Summary AVL Trees Requires a little more work for insertion and deletion But, since trees are mostly used for searching More work for insert and delete is worth the performance gain for searching
- 33. Self-adjusting Structures Consider the following AVL Tree 44 17 78 32 50 88 48 62
- 34. Self-adjusting Structures Consider the following AVL Tree 44 17 78 32 50 88 48 62 Suppose we want to search for the following sequence of elements: 48, 48, 48, 48, 50, 50, 50, 50, 50.
- 35. Self-adjusting Structures Consider the following AVL Tree 44 17 78 32 50 88 48 62 Suppose we want to search for the following sequence of elements: 48, 48, 48, 48, 50, 50, 50, 50, 50. In this case, is this a good structure?
- 36. Self-adjusting Structures Splay Trees (Tarjan and Sleator 1985) Binary search tree. Every accessed node is brought to the root Adapt to the access probability distribution
- 37. Practice time! A) 50 10 80 30 60 10 30 42 80 50 60 10 30 50 80 42 60 10 30 50 80 42 60 42 10 30 80 60 50 B) C) D) Which of the following is the updated AVL tree after inserting 42? Starting with this AVL tree: