Chien-Pin Hsu

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AVL Trees

• Presented by
• Chien-Pin Hsu
• CS146
• Prof. Sin-Min Lee

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Outline

• Introduction
• Definitions
• Height-Balanced Trees
• Picking Out AVL Trees
• Why AVL Trees?
• Added Complexity
• Violations
• The Balance Factor
• What is a rotation?
• Single Rotations
• Double Rotations

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Introduction

• AVL tree is the first balanced binary search tree
  (name after its discovers, Adelson-Velskii and
  Landis).
• The AVL tree is a balanced search tree that has
  an additional balance condition.
• The balance condition must be easy to maintain
  and ensures that the depth of the tree is O(logN)

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Definitions

• An AVL tree is a binary search tree that is
  height balanced meaning that it has these two
  properties..
• 1. The sub-tree of every node must differ
  in height by at most one.
• 2. Every sub-tree is an AVL tree

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Height-Balanced Trees

• Height of a tree is the length of the longest
  path from a root to a leaf.
• A binary search tree T is a height-balanced
  k-tree or HBk-tree, if each node in the tree
  has the HBk property.
• A node has the HBk property if the height of
  the left and right sub-trees of the node differ
  in height by at most k.
• A HB1 tree is called an AVL tree.

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Picking Out AVL Trees

• Notation NODE Left-Ht, Right-Ht

Yes! 5 2, 1 4 1, 0 3 0, 0 7 0, 0
No! 10 3, 1
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Picking Out AVL Trees (Contd)

• Notation NODE Left-Ht, Right-Ht

No! 8 4, 2 4 3, 1 3 2, 0
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Why AVL Trees?

• The worst case for Binary Search Trees
• Insert O(n).
• Delete O(n).
• Search O(n).
• The worst case for AVL trees
• Insert O(log n).
• Delete O(log n).
• SearchO(log n).
• Recall when comparing the time complexity
• O(log n) lt O(n)

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Why AVL Trees? (Contd)

• AVL trees allow for logarithmic Inserts and
  Deletes, and they allow for easy traversals, such
  as the In-order traversal if we want to sort
  our elements.

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Added Complexity

• Although we have made the Insertion, Deletion
  Searching better, an additional amount of
  complexity does arise during the Insertions and
  the Deletions.
• After Inserting and Deleting for an AVL tree, the
  new tree might violate the two properties of the
  AVL tree.

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Violations
Notation NODE Left-Ht, Right-Ht
No! 5 3,1 4 2,0
No! 5 2, 0
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The Balance Factor

• In order to detect when a violation of a AVL
  tree occurs, we need to have each node keep track
  of the difference in height between its right and
  left sub-trees.
• Notation for the balance factor
• bf (node) Left-Ht Right- Ht
• To be a valid AVL Tree, -1 lt bf(x) lt 1 for all
  nodes.

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The Balance Factor (Contd)

• Notation for the balance factor
• bf (node) Left-Ht Right- Ht

bf(8) 3 3 0 bf(4) 2 1 1 bf(10) 1
2 -1 bf(3) 1 0 1 bf(5) 0 0 0 bf(9)
0 0 0 bf(11) 0 1 -1 bf(1) 0 0
0 bf(12) 0 0 0
Valid Tree!!
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The Balance Factor (Contd)
Notation for the balance factor bf
(node) Left-Ht Right- Ht
bf(8) 2 0 2 bf(4) 1 0 1 Bf(3) 0 0
0 Invalid Tree!!
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What is a rotation?

• A rotation is an operation to rearrange nodes to
  maintain balance of an AVL tree after adding and
  removing a node
• There are two case to perform the rotation
• 1) Single Rotation
• 2) Double Rotation
• When to perform rotations?
• we need to perform a rotation anytime a nodes
  balance factor is 2 or -2

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Single Rotations (Contd)

• Algorithm for right rotation
• Algorithm rotateRight(nodeN) nodeC left
  child of nodeN Set nodeNs left child to
  nodeCs right child Set nodeCs right child to
  nodeN

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Single Rotations

• After inserting (a) 60 (b) 50 (c) 20 into an
  initially empty binary search tree, the tree is
  not balanced d) a corresponding AVL tree rotates
  its nodes to restore balance.

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Single Rotations (Contd)

• Algorithm for left rotation
• Algorithm rotateLeft(nodeN) nodeC right
  child of nodeN Set nodeNs right child to
  nodeCs left child Set nodeCs left child to
  nodeN

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Single Rotations (Contd)
(a) Adding 80 to tree does not change the
balance of the tree (b) a subsequent
addition of 90 makes tree unbalanced (c)
left rotation restores balance.
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Double Rotations

• Before and after an addition to an AVL sub-tree
  that requires both a right rotation and a left
  rotation to maintain its balance.

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Double Rotations (Contd)
Before and after an addition to an AVL sub-tree
that requires both a left and right rotation to
maintain its balance.
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Double Rotations (Contd)

• Algorithm rotateLeftRight(nodeN)
• nodeC left child of nodeNSet nodeNs left
  child to the sub-tree produced by
• rotateLeft(nodeC) rotateRight(nodeN)

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Double Rotations (Contd)

1. Inserting 10 ,40,and 55 maintain its balance
2. Inserting 35 destroys the balance
3. After a left rotation
4. After a right rotation

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Double Rotations (Contd)

• Algorithm rotateRightLeft(nodeN)
• nodeC right child of nodeNSet nodeNs
  right child to the sub-tree produced by
• rotateRight(nodeC) rotateLeft(nodeN)

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Double Rotations (Contd)
(a) Adding 70 to the tree destroy its balance.
To restore the balance, perform both (b) a
right rotation and (c) a left rotation.
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The End