Lec-13 AVL Trees

8/3/2019 Lec-13 AVL Trees

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AVL Trees


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AVL TreesFor some binary search tree, the heightcould be quite large, which is not good

for searching.

An AVL tree is a binary search tree witha balance condition

To ensure that the height of the tree is

O(logN)For every node in the AVL tree, the height

of left subtree and the height of the right

subtree can differ by at most one.


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AVL Trees

The height of the left subtree minus the

height of the right subtree of a node is

called the balance of the node. For an

AVL tree, the balances of the nodes arealways -1, 0 or 1.

The height of an empty tree is defined to

be -1.Given an AVL tree, if insertions or

deletions are performed, the AVL tree

may notremain height balanced.


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AVL Trees

5

2 8

1 4

3

7

An AVL Tree

7

2 8

1 4

3 5

Not an AVL Tree


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AVL Trees

To maintain the height balancedproperty of the AVL tree after insertionor deletion, it is necessary to perform a

transformation on the tree so that(1) the inorder traversal of thetransformed tree is the same as for theoriginal tree (i.e., the new tree remains

a binary search tree).(2) the tree after transformation is heightbalanced.


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After an insertion, only nodes that are on

the path from the insertion point to the

root might have their balance altered

Follow the path up to the root, find the

first node (i.e., deepest) whose new

balance violates the AVL condition. Call

this node a.

Rebalance the tree at node a.

This guarantees that the entire tree is

balanced.

AVL Trees


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AVL Trees

Violation may occur when an

insertion into

1. left subtree of left child ofa (LL case)

2. right subtree of left child ofa (LR

case)

3. left subtree of right child ofa (RL

case)

4. right subtree of right child ofa (RR

case)


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AVL Trees

Rotation

- to restore the AVL tree after

insertion- single rotation

For case 1 and 4

- double rotationFor case 2 and 3


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AVL Trees: Single Rotation

Example: 3 2 1 4 5 6 7

construct binary search tree without

height balanced restriction

depth of tree = 4

i.e. LL casei.e. LL case


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Construct AVL tree (height balanced)


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Insert 4, 5Insert 4, 5

Insert 6Insert 6


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Insert 7Insert 7


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AVL Trees: Double Rotation

Single rotation fails to fix cases 2 and 3

(i.e. LR and RL cases)


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case 3 (RL case)


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Example:


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Insert 14


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Insert 13 (This is single rotation: RR

Case)


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Insert 12


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Insert 11 and 10 (single rotation), then 8


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Inserting 9

Lec-13 AVL Trees

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