This document presents information about AVL trees through a slideshow presentation. It begins by defining AVL trees as binary search trees where the heights of the subtrees of any node differ by at most 1. It then discusses that AVL trees ensure searching, insertion, and deletion take O(log N) time by keeping the tree balanced through rotations. The presentation notes that AVL trees remain balanced through single and double rotations during insertions and deletions, and that rebalancing can be done in O(log N) time. It concludes by emphasizing that AVL trees guarantee O(log N) search time in dynamic environments through maintaining balance.

1. Presentation On :
AVL TREES
Group Member:
Chhatra Thapa (Sid:10935)
Susat Maharjan (Sid:10954)
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2. AVL TREE IS…
• named after Adelson-Velskii and Landis
• the first dynamically balanced trees to be propose
• Binary search tree with balance condition in which the sub-trees of
each node can differ by at most 1 in their height
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3. DEFINITION OF A BALANCED TREE
• Ensure the depth = O(log N)
• Take O(log N) time for searching, insertion, and deletion
• Every node must have left & right sub-trees of the same height
• Goal: keep the height of a binary search tree O(log N).
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4. AN AVL TREE HAS THE FOLLOWING PROPERTIES:
• Sub-trees of each node can differ by at most 1 in their height
• Every sub-trees is an AVL tree
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5. AVL TREE?
YES NO
Each left sub-tree has Left sub-tree has height 3,
height 1 greater than but right sub-tree has
each right sub-tree height 1
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6. INSERTION AND DELETIONS
• It is performed as in binary search trees
• If the balance is destroyed, rotation(s) is performed to correct
balance
• For insertions, one rotation is sufficient
• For deletions, O(log n) rotations at most are needed
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7. Single Rotation
left sub-tree is one level move ① up a level and
deeper than the right sub-tree ② down a level
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8. Single rotation : Rotate right
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9. Single rotation : Rotate left
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10. Double Rotation
Left sub-tree is two level Move ② up two levels and
deeper than the right sub-tree ③ down a level
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11. Insertion
Insert 6
Imbalance at 8
Perform rotation with 7
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13. Deletion
Delete 4
Imbalance at 3
Perform rotation with 2
Imbalance at 5
Perform rotation with 8 13

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15. KEY POINTS
• AVL tree remain balanced by applying rotations, therefore it
guarantees O(log N) search time in a dynamic environment
• Tree can be re-balanced in at most O(log N) time
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16. THANK YOU

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