Balanced Trees. Maintaining Balance Binary Search Tree – Height governed by Initial order Sequence of insertion/deletion – Changes occur at leaf nodes.

Balanced TreesBalanced Trees

Maintaining BalanceMaintaining 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”)

2-3 Tree Nodes2-3 Tree Nodes

> L< S

S

> S< S>S <L

S L

BST InsertionBST Insertion60

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2-3 Tree Insertion (36)2-3 Tree Insertion (36)

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2-3 Tree Insertion (36)2-3 Tree Insertion (36)

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2-3-4 Trees, Red-Black 2-3-4 Trees, Red-Black TreesTrees

2-3 Tree Insertion (38 – 2-3 Tree Insertion (38 – Leaf Node)Leaf Node)30

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Generalized Insertion – Generalized Insertion – Leaf NodeLeaf Node

P

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S eL

2-3 Tree Insertion (36 2-3 Tree Insertion (36 Internal Node)Internal Node)

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2-3 Tree Insertion (36)2-3 Tree Insertion (36)

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Generalized InsertionGeneralized Insertion,, Internal NodeInternal Node

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2-3-4 Trees2-3-4 Trees If allowing

– 2 values/node– 3 children/node is better…

What about– 3 values/node– 4 children/node ???

Definition

S M L

> L< S >S <M >M <L

>L<S >S<L

S L

S

>S<S

–See 2-3 Tree

“3-Node”

…AND…

Insertion in 2-3-4 TreeInsertion 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?

Insertion in 2-3-4 Tree (20)Insertion in 2-3-4 Tree (20)

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Insertion in 2-3-4 Tree (70)Insertion in 2-3-4 Tree (70)

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Generalized Insertion in 2-Generalized Insertion in 2-3-4 Tree3-4 Tree

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Splitting a 4-Node with a 2-Node Parent

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Generalized Insertion in 2-Generalized Insertion in 2-3-4 Tree3-4 Tree

Splitting a 4-Node with a 3-Node Parent

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cba d

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Binary Search Tree Binary Search Tree RepresentationRepresentation

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

B

A C

S T U V

Red-Black Tree Red-Black Tree RepresentationRepresentation

A B C

S T U V

4-Node

Red-Black Tree Red-Black Tree RepresentationRepresentation

A B

S T U

B

A U

S T

A

S B

T U

3-Node

Red-Black Tree TranslationRed-Black Tree Translation10

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Red-Black Tree InsertionRed-Black Tree Insertion

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Parent (root) is 4-NodeB

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Red-Black Tree InsertionRed-Black Tree InsertionParent is 2-Node

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Red-Black Tree InsertionRed-Black Tree InsertionParent is 3-Node (version 1 of 2)

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Red-Black Tree InsertionRed-Black Tree InsertionParent is 3-Node (version 2 of 2) - Rotation

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Adelson-Velskii and Landis Adelson-Velskii and Landis (AVL)(AVL)

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AVL TreeAVL Tree 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 x is -1, 0, or 1

The height of an AVL tree that has n nodes is at most 1.44 log2 (n+2).

The height of every n node binary tree is at least log2 (n+1).

0 0

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AVL Search TreeAVL Search Tree

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insert(9)insert(9)

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insert(29)insert(29)

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RR imbalance => new node is in right subtree of right subtree of blue node (node with bf = -2)

insert(29)insert(29)

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AVL RotationsAVL 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

LL Rotation AlgorithmLL Rotation AlgorithmBinaryNode LL_Rotate ( BinaryNode k2) {

BinaryNode k1 = k2.left;k2.left = k1.right;k1.right = k2;return k1;

}

RR Rotation AlgorithmRR Rotation AlgorithmBinaryNode RR_Rotate ( BinaryNode k1 ) {

BinaryNode k2 = k1.right;k1.right = k2.left;k2.left = k1;return k2;

}

Single Rotation Running Single Rotation Running ExampleExample

Consider the formation of an AVL tree by inserting the keys 1, 2, 3, 4, 5, 6 and 7 sequentially

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Single Rotation (2)Single Rotation (2)

Consider the formation of an AVL tree by inserting the keys 1, 2, 3, 4, 5, 6 and 7 sequentially

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Why Double Rotation?Why Double Rotation?Single Rotation cannot solve LR or RL

LR Double RotationLR Double RotationBinaryNode LR_doubleRotate ( BinaryNode k3 ){

k3.left = RR_Rotate ( k3.left );return LL_Rotate ( k3 );

}

LR Double Rotation LR Double Rotation Example Example

RL Double RotationRL Double RotationBinaryNode RL_doubleRotate ( BinaryNode k1 ) {

k1.right = LL_Rotate ( k1.right );return RR_Rotate ( k1 );

}

More Double Rotation More Double Rotation ExampleExample

Consider the insertion of 15, 14, 13 into the AVL tree built by inserting 1, ...,7 sequentially

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Double Rotation Example Double Rotation Example (2)(2)

Consider the insertion of 15, ..., 10, 9, 8, and 8.5 into the AVL tree built by inserting 1, ...,7 sequentially

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B-TreesB-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

B-treeB-tree A B-tree of order m has the following properties: The data items are stored at the leaves in order The root is either a leaf or has 2 to m children All non-leaf nodes (except the root) have m/2 to m children Each non-leaf node has at most m-1 keys: key i is the data value

in subtree i+1 All leaves are at the same depth and have L/2 to L items

B-tree Insertion - SplitB-tree Insertion - Split L and M are determined by the sizes of a node, the size of an item

and the size of a key Requiring nodes to be half full guarantees that the B-tree will be

pretty balanced and will not degenerate into a simple binary tree, or even into a linked list

A B-tree of order 3 is also known as a 2-3 tree Insertion of 55 in the example B-tree causes a split into two leaves

B-tree Insertion - B-tree Insertion - Cascaded SplitCascaded Split Further insertion of 40 in the B-tree causes a split into

two leaves and then a split of the parent node

B-tree DeletionB-tree Deletion Deletion is done similarly If the number of items in a leaf falls below the

minimum, adopt an item from a neighboring leaf If the number of items in the neighboring leaves are

also minimum, combine two leaves. Their parent will then lose a child and it may need to be combined with its neighbor

This combination process should be recursively executed up the tree until:– Getting to the root– A parent has more than the minimum number of children