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Balanced Search Trees (partial) Chapter 29 Slides by Steve Armstrong LeTourneau University Longview, TX  2007,  Prentice Hall - Edited by Nadia Al-Ghreimil.

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Presentation on theme: "Balanced Search Trees (partial) Chapter 29 Slides by Steve Armstrong LeTourneau University Longview, TX  2007,  Prentice Hall - Edited by Nadia Al-Ghreimil."— Presentation transcript:

1 Balanced Search Trees (partial) Chapter 29 Slides by Steve Armstrong LeTourneau University Longview, TX  2007,  Prentice Hall - Edited by Nadia Al-Ghreimil 2009

2 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Chapter Contents AVL Trees  Single Rotations  Double Rotations  Implementation Details B-Trees

3 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X AVL Trees An AVL tree is a binary search tree Whenever it becomes unbalanced  Rearranges its nodes to get a balanced binary search tree Balance of a binary search tree upset when  Add a node  Remove a node Thus during these operations, the AVL tree must rearrange nodes to maintain balance

4 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X AVL Trees Fig. 29-1 After inserting (a) 60; (b) 50; and (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.

5 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X AVL Trees Fig. 29-2 (a) Adding 80 to tree in Fig. 29-1d does not change the balance of the tree; (b) a subsequent addition of 90 makes tree unbalanced; (c) left rotation restores balance.

6 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Single Rotations N C T1 T2 T3 new T1 h +1

7 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Single Rotations N C T2 T3 T1 h +1 T2 N T3 C T1 h +1

8 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Single Rotations N C T2 T3 T1 h +1 T2 N T3 C T1 h

9 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Single Rotations Fig 28-4 Before and after a right rotation restores balance to an AVL tree

10 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Single Rotations Fig 28-5 Before and after an addition to an AVL subtree that requires a left rotation to maintain balance.

11 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Single Rotations Algorithm for right rotation

12 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Single Rotations Algorithm for left rotation

13 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Double Rotations Fig. 29-6 (a) Adding 70 to the tree in Fig. 29-2c destroys its balance; to restore the balance, perform both (b) a right rotation and (c) a left rotation.

14 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Double Rotations Fig. 29-7 (a-b) Before and after an addition to an AVL subtree that requires both a right rotation and a left rotation to maintain its balance.

15 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Double Rotations Fig. 29-7 (c-d) Before and after an addition to an AVL subtree that requires both a right rotation and a left rotation to maintain its balance.

16 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Double Rotations Algorithm to perform the right-left double rotation illustrated in Fig. 29-7

17 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Double Rotations Fig. 29-8 (a) The AVL tree in Fig. 29-6 after additions that maintain its balance; (b) after an addition that destroys the balance … continued →

18 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Double Rotations Fig. 29-7 (ctd.) (c) after a left rotation; (d) after a right rotation.

19 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Double Rotations Fig. 29-9 Before and after an addition to an AVL subtree that requires both a left and right rotation to maintain its balance.

20 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Double Rotations Algorithm to perform the left-right double rotation illustrated in Fig. 29-9

21 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Double Rotations An imbalance at node N of an AVL tree can be corrected by a double rotation if  The addition occurred in the left subtree of N's right child (left-right rotation) or …  The addition occurred in the right subtree of N's left child (right-left rotation) A double rotation is accomplished by performing two single rotations  A rotation about N's grandchild  A rotation about node N's new child

22 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Double Rotations Fig. 29-10 The result of adding 60, 50, 20, 80, 90, 70, 55, 10, 40, and 35 to an initially empty (a) AVL tree; (b) binary search tree. Click to view outline of class AVLTree

23 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X B-Trees B-tree of order m  Balanced multiway search tree of order m Properties for maintaining balance  Root has either no children or between 2 and m children  Other interior nodes (nonleaves) have between m/2 and m children  All leaves are on the same level High order B-tree works well when tree maintained in external (disk) storage.

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