Balanced Binary Search Trees
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1 Balanced Binary Search Trees Why is our balance assumption so important? Lets look at what happens if we insert the following numbers in order without rebalancing the tree: Pearson Addison-Wesley. All rights reserved. 1-45
2 A degenerate binary tree Pearson Addison-Wesley. All rights reserved. 1-46
3 Degenerate Binary Trees The resulting tree is called a degenerate binary tree Degenerate binary search trees are far less efficient than balanced binary search trees (O(n) on find as opposed to O(logn)) Pearson Addison-Wesley. All rights reserved. 1-47
4 Balancing Binary Trees There are many approaches to balancing binary trees One method is brute force Write an inorder traversal to a file Use a recursive binary search of the file to rebuild the tree Pearson Addison-Wesley. All rights reserved. 1-48
5 Balancing Binary Trees Better solutions involve algorithms such as red-black trees and AVL trees that persistently maintain the balance of the tree Most all of these algorithms make use of rotations to balance the tree Lets examine each of the possible rotations Pearson Addison-Wesley. All rights reserved. 1-49
6 Right Rotation Right rotation will solve an imbalance if it is caused by a long path in the left sub-tree of the left child of the root Make the left child element of the root the new root element. Make the former root element the right child element of the new root. Make the right child of what was the left child of the former root the new left child of the former root Pearson Addison-Wesley. All rights reserved. 1-50
7 Unbalanced tree and balanced tree after a right rotation Pearson Addison-Wesley. All rights reserved. 1-51
8 Left Rotation Left rotation will solve an imbalance if it is caused by a long path in the right sub-tree of the right child of the root Make the right child element of the root the new root element. Make the former root element the left child element of the new root. Make the left child of what was the right child of the former root the new right child of the former root Pearson Addison-Wesley. All rights reserved
9 Unbalanced tree and balanced tree after a left rotation Pearson Addison-Wesley. All rights reserved. 1-53
10 Rightleft Rotation Rightleft rotation will solve an imbalance if it is caused by a long path in the left sub-tree of the right child of the root Perform a right rotation of the left child of the right child of the root around the right child of the root, and then perform a left rotation of the resulting right child of the root around the root Pearson Addison-Wesley. All rights reserved. 1-54
11 A rightleft rotation Pearson Addison-Wesley. All rights reserved. 1-55
12 Leftright Rotation Leftright rotation will solve an imbalance if it is caused by a long path in the right sub-tree of the left child of the root Perform a left rotation of the right child of the left child of the root around the left child of the root, and then perform a right rotation of the resulting left child of the root around the root Pearson Addison-Wesley. All rights reserved. 1-56
13 A leftright rotation Pearson Addison-Wesley. All rights reserved. 1-57
14 AVL Trees AVL trees keep track of the difference in height between the right and left sub-trees for each node This difference is called the balance factor If the balance factor of any node is less than -1 or greater than 1, then that sub-tree needs to be rebalanced The balance factor of any node can only be changed through either insertion or deletion of nodes in the tree Pearson Addison-Wesley. All rights reserved. 1-58
15 AVL Trees If the balance factor of a node is -2, this means the left sub-tree has a path that is too long If the balance factor of the left child is -1, this means that the long path is the left sub-tree of the left child In this case, a simple right rotation of the left child around the original node will solve the imbalance Pearson Addison-Wesley. All rights reserved. 1-59
16 A right rotation in an AVL tree Pearson Addison-Wesley. All rights reserved. 1-60
17 AVL Trees If the balance factor of a node is +2, this means the right sub-tree has a path that is too long Then if the balance factor of the right child is +1, this means that the long path is the right sub-tree of the right child In this case, a simple left rotation of the right child around the original node will solve the imbalance Pearson Addison-Wesley. All rights reserved. 1-61
18 AVL Trees If the balance factor of a node is +2, this means the right sub-tree has a path that is too long Then if the balance factor of the right child is -1, this means that the long path is the left sub-tree of the right child In this case, a rightleft double rotation will solve the imbalance Pearson Addison-Wesley. All rights reserved. 1-62
19 A rightleft rotation in an AVL tree Pearson Addison-Wesley. All rights reserved. 1-63
20 AVL Trees If the balance factor of a node is -2, this means the left sub-tree has a path that is too long Then if the balance factor of the left child is +1, this means that the long path is the right sub-tree of the left child In this case, a leftright double rotation will solve the imbalance Pearson Addison-Wesley. All rights reserved. 1-64
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