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Balanced Trees

Balanced Trees. Maintaining Balance. Binary Search Tree Height governed by Initial order Sequence of insertion/deletion Changes occur at leaf nodes Need a structure that tends to maintain balance How? Grow in ‘width’ first, then height Accommodate horizontal growth

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Balanced Trees

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  1. Balanced Trees

  2. Maintaining Balance • Binary Search Tree • Height governed by • Initial order • Sequence of insertion/deletion • Changes occur at leaf nodes • Need a structure that tends to maintain balance • How? • Grow in ‘width’ first, then height • Accommodate horizontal growth • More data at each level • Nodes of two forms • One data member and two children (“Two node”) • Two data members and three children (“Three node”)

  3. S < S > S 2-3 Tree Nodes S L < S >S <L > L

  4. 60 60 60 30 30 90 90 30 90 10 10 50 50 80 80 100 100 10 50 80 100 20 20 40 40 70 70 20 40 70 39 39 BST Insertion 39

  5. 50 30 70 90 50 10 20 40 60 80 100 30 70 90 10 20 39 40 60 80 100 2-3 Tree Insertion (39)

  6. 30 30 10 20 39 40 10 20 38 39 40 30 39 10 20 38 40 2-3 Tree Insertion (38)

  7. 50 30 39 70 90 50 10 20 38 40 60 80 100 30 39 70 90 10 20 37 38 40 60 80 100 2-3 Tree Insertion (37)

  8. 50 70 90 60 80 100 30 39 10 20 37 38 40 2-3 Tree Insertion (36)

  9. 30 39 10 20 36 37 38 40 37 50 50 30 39 30 37 39 30 39 70 90 10 20 37 38 40 10 20 36 38 40 10 20 36 38 40 60 80 100 2-3 Tree Insertion (36)

  10. 2-3-4 Trees, Red-Black Trees

  11. 30 30 10 20 39 40 10 20 38 39 40 30 39 10 20 38 40 2-3 Tree Insertion (38 – Leaf Node)

  12. P M P S M L e S L e Generalized Insertion – Leaf Node

  13. 50 70 90 60 80 100 30 39 10 20 37 38 40 2-3 Tree Insertion (36 Internal Node)

  14. 30 39 10 20 36 37 38 40 37 50 50 30 39 30 37 39 30 39 70 90 10 20 37 38 40 10 20 36 38 40 10 20 36 38 40 60 80 100 2-3 Tree Insertion (36)

  15. P M P S M L e S L e a a b b c c d d Generalized Insertion, Internal Node

  16. S <S >S<L >L S L <S >S S M L < S >S <M >M <L > L 2-3-4 Trees • If allowing • 2 values/node • 3 children/node is better… • What about • 3 values/node • 4 children/node ??? • Definition • See 2-3 Tree • …AND… • “3-Node”

  17. Insertion in 2-3-4 Tree • Same scheme as 2-3 Tree • Traverse down tree • If node is 4-node • Split tree • Insert into new tree • This means what in terms of… • Growth in Height versus Width? • How far ‘up’ will insertion cascade?

  18. 30 30 10 20 60 10 60 Insertion in 2-3-4 Tree (20) 20 10 30 60

  19. 30 30 50 10 20 40 50 60 10 20 40 60 30 50 10 20 40 60 70 Insertion in 2-3-4 Tree (70) 70

  20. P M P S M L e S L e a a a b b b c c c d d d P S M L e Generalized Insertion in 2-3-4 Tree Splitting a 4-Node with a 2-Node Parent

  21. P Q M P Q S M L e f S L e f a a a b b b c c c d d d P Q S M L e f Generalized Insertion in 2-3-4 Tree Splitting a 4-Node with a 3-Node Parent

  22. Binary Search Tree Representation • Color-code node-type information • 2-Node • Ordinary binary search tree node • 4-Node with values A, B, and C • Center B is Black • Children A and C are Red • 3-Node with values A, and B (two choices) • A is Black parent, B is Red right child • or – • B is Black parent, A is Red left child • Ignore color-code • Structure of binary search tree • Two ways to change tree • Color change • Rotations

  23. B A B C A C S T U V S T U V Red-Black Tree Representation 4-Node

  24. A A B S B S T U T U B A U S T Red-Black Tree Representation 3-Node

  25. 10 8 10 20 8 20 3 9 12 40 1 3 4 9 12 15 30 40 1 4 15 30 Red-Black Tree Translation

  26. B A B C B A C S T U V A C S T U V S T U B V A C S T U V Red-Black Tree Insertion Parent (root) is 4-Node 2-3-4 Representation Red-Black Representation

  27. P M P S M L e S L e a a b b c c d d P P M E M E S L S L A B C D A B C D Red-Black Tree Insertion Parent is 2-Node 2-3-4 Representation Red-Black Representation

  28. M P Q S L e f a a b b c c d d P Q P P M Q M Q S M L e f S L S L E F E F A B C D A B C D Red-Black Tree Insertion Parent is 3-Node (version 1 of 2) 2-3-4 Representation Red-Black Representation

  29. Q F P a b c d E M P Q S L P A B C D M Q S M L e f S L E F A B C D Red-Black Tree Insertion Parent is 3-Node (version 2 of 2) - Rotation

  30. 30 20 20 20 10 30 10 30 10 40 Adelson-Velskii and Landis (AVL)

  31. AVL Tree -1 • Binary tree • For every node x, define its balance factor balance factor of x = height of left subtree of x - height of right subtree of x • Balance factor of every node xis -1, 0, or 1 • The height of an AVL tree that has nnodes is at most1.44 log2 (n+2). • The height of every nnode binarytree is at leastlog2 (n+1). 1 1 -1 0 1 0 0 -1 0 0 0 0

  32. -1 10 1 1 7 40 -1 0 1 0 45 3 8 30 0 -1 0 0 0 60 35 1 20 5 0 25 AVL Search Tree

  33. insert(9) -1 10 0 1 1 7 40 -1 0 1 -1 0 45 3 8 30 0 -1 0 0 0 0 60 35 1 9 20 5 0 25

  34. insert(29) -1 10 1 1 7 40 -1 0 1 0 45 3 8 30 0 -1 0 0 0 -2 60 35 1 20 5 0 -1 RR imbalance => new node is in right subtree of right subtree of blue node (node with bf = -2) 25 0 29

  35. insert(29) -1 10 1 1 7 40 -1 0 1 0 45 3 8 30 0 0 0 0 0 60 35 1 25 5 0 0 20 29 RR rotation.

  36. AVL Rotations • Single Rotation • RR • LL • Double Rotation • RL • new node in left subtree of right subtree • Go right down and then go left down • LR

  37. LL Rotation Algorithm BinaryNode LL_Rotate ( BinaryNode k2) {BinaryNode k1 = k2.left;k2.left = k1.right;k1.right = k2;return k1; }

  38. RR Rotation Algorithm BinaryNode RR_Rotate ( BinaryNode k1 ) {BinaryNode k2 = k1.right;k1.right = k2.left;k2.left = k1;return k2; }

  39. Single Rotation Running Example Consider the formation of an AVL tree by inserting the keys 1, 2, 3, 4, 5, 6 and 7 sequentially 0 -2 1 1 2 1 -1 0 0 2 2 3 1 2 0 2 3 3 1 4 4 1 5 3 5

  40. Single Rotation (2) Consider the formation of an AVL tree by inserting the keys 1, 2, 3, 4, 5, 6 and 7 sequentially 4 2 -2 2 5 4 1 1 6 3 5 3 6

  41. Why Double Rotation? • Single Rotation cannot solve LR or RL

  42. LR Double Rotation BinaryNode LR_doubleRotate ( BinaryNode k3 ){k3.left = RR_Rotate ( k3.left );return LL_Rotate ( k3 ); }

  43. LR Double Rotation Example

  44. RL Double Rotation BinaryNode RL_doubleRotate ( BinaryNode k1 ) {k1.right = LL_Rotate ( k1.right );return RR_Rotate ( k1 ); }

  45. More Double Rotation Example • Consider the insertion of 15, 14, 13 into the AVL tree built by inserting 1, ...,7 sequentially 4 4 2 2 6 6 1 7 1 14 3 5 3 5 7 15 15 14

  46. Double Rotation Example (2) • Consider the insertion of 15, ..., 10, 9, 8, and 8.5 into the AVL tree built by inserting 1, ...,7 sequentially 4 4 6 7 2 2 14 5 14 6 1 3 1 3 7 15 13 15 5 13

  47. 20 40 10 40 20 50 30 50 10 30 60 60 Single Left Rotation

  48. 40 40 30 50 20 50 20 35 60 10 30 60 10 25 25 35 22 22 Double Rotation (Step One)

  49. 40 30 30 50 20 40 20 35 60 10 25 35 50 10 25 22 60 22 Double Rotation (Step Two)

  50. B-Trees • Binary tree is still not efficient enough for searching • A perfectly balanced binary tree has 5 levels for 25 nodes, while a 5-ary tree has only 3 levels • Especially useful for external usage

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