lecture7 data structure(tree)

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Tree(Data Structure)

Lecture 7

Abdisalam Issa-Salwe

Taibah University College of Computer Science & Engineering

Computer Science Department

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Outline

� Binary trees

� Binary search trees

� AVL trees

� B trees

� Applications

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3

Tree ADT

A tree is a collection (may be empty) of nodes, containing:

� a distinguished node called the root r,

� zero or more non-empty subtrees T1, T2, …, Tk,

� A directed edge from the root r to the root of each

subtree. root

T1

T2

Tk

…

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Terminologies

rootparent

children

grandchildren

siblings

subtrees

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5

Terminologies

length of path

path

ancestor

length of path from the root

descendant

depth

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Terminologies

height

4

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Tree: Implementationclass TreeNode

{

Object element;

TreeNode firstChild;

TreeNode nextSibling;

}

nextsibling

nextSibling=null

nextSibling=null

nextSibling=null

firstChild=nullfirstChild=null

firstChild=nullfirstChild=null

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Binary Trees

� A tree in which no node can have more

than 2 children.

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9

Binary Tree: Implementation

Class BinaryNode

{

Object element;

BinaryNode left;

BinaryNode right;

}

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node.element=9node.left=nullnode.right=null

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node.element=5node.left=node node.right=null

3

node.element=3node.left=node node.right=node

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Binary Tree: Implementation

class BinaryNode

{ // Constructors

BinaryNode ( Comparable theElement )

{ this ( theElement, null, null); }

BinaryNode

(Comparable theElement, BinaryNode lt, BinaryNode rt)

{ element = theElement;

left = lt;

right = rt; }

// Friendly data; accessible by other package routines

Comparable element; // The data in the node

BinaryNode left; // Left child

BinaryNode right; // Right child

}

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Binary Search Trees

Properties of a binary search tree T

1. T is a binary tree.

2. For each node n in T, whose left subtree

is Tl and right subtree is Tr,

• the item in each node in Tl is smaller than the item in n.

• the item in each node in Tr is larger than the item in n.

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Example

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3

42

61

75

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129

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Binary Search Trees

public class BinarySearchTreepublic class BinarySearchTreepublic class BinarySearchTreepublic class BinarySearchTree

{{{{ public BinarySearchTree( )public BinarySearchTree( )public BinarySearchTree( )public BinarySearchTree( )

{ root = null;{ root = null;{ root = null;{ root = null; }}}}

public void insert( Comparable x )public void insert( Comparable x )public void insert( Comparable x )public void insert( Comparable x )

{ root = insert( x, root ); { root = insert( x, root ); { root = insert( x, root ); { root = insert( x, root ); }}}}

public void remove( Comparable x )public void remove( Comparable x )public void remove( Comparable x )public void remove( Comparable x )

{ root = remove( x, root ); { root = remove( x, root ); { root = remove( x, root ); { root = remove( x, root ); }}}}

public Comparable find( Comparable x )public Comparable find( Comparable x )public Comparable find( Comparable x )public Comparable find( Comparable x )

{ return elementAt( find( x, root ) ); }{ return elementAt( find( x, root ) ); }{ return elementAt( find( x, root ) ); }{ return elementAt( find( x, root ) ); }

public void makeEmpty( )public void makeEmpty( )public void makeEmpty( )public void makeEmpty( )

{ root = null; { root = null; { root = null; { root = null; }}}}

............

private BinaryNode root;private BinaryNode root;private BinaryNode root;private BinaryNode root; }}}}

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FIND

8

3

42

61

75

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129

Find 6<

>

>

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Method find

private BinaryNode find

( Comparable x, BinaryNode t )

{

if( t == null ) return null;

if( x.compareTo( t.element ) < 0 )

return find( x, t.left );

else if( x.compareTo( t.element ) > 0 )return find( x, t.right );

else return t; // Match

}

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INSERT

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3

42

61

75

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Insert 10

10

>

<

10

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Method insert

private BinaryNode insert

( Comparable x, BinaryNode t )

{ if( t == null )

t = new BinaryNode( x, null, null );

else if( x.compareTo( t.element ) < 0 )

t.left = insert( x, t.left );

else if( x.compareTo( t.element ) > 0 )

t.right = insert( x, t.right );

else ; // Duplicate; do nothing

return t;

}

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FindMax, FindMin

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3

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61

75

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129

min

max

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Methods findMin & findMax

private BinaryNode findMin( BinaryNode t )

{ if( t == null )

return null;

else if( t.left == null )

return t;

return findMin( t.left );

}

private BinaryNode findMax( BinaryNode t )

{ if( t != null )

while( t.right != null )

t = t.right;

return t;

}

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REMOVE

11

3

72

51

64

13

1412

Remove 7

9

10

11

21

REMOVE

11

3

72

1

13

1412

Remove 7

9

108

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REMOVE

11

3

72

1

13

1412

Remove 7

5

64

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Method Removeprivate BinaryNode remove(Comparable x,BinaryNode t)

{ if(t == null) return t; // Item not found;do nothing

if( x.compareTo(t.element) < 0 )

t.left = remove(x, t.left);

else if ( x.compareTo(t.element) > 0 )

t.right = remove(x, t.right);

else if (t.left!=null && t.right!=null) // 2 child

{ t.element = findMin(t.right).element;

t.right = remove(t.element, t.right);

}

else

t = (t.left != null) ? t.left : t.right;

return t;

}

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AVL Trees

� An AVL tree is a binary search tree with

a balance condition. A self-balancing

binary search tree.

� AVL tree is named after G.M. Adelson-

Velskii and E.M. Landis.

Balance condition

� For every node in the tree, the height of

the left & right subtrees can differ by at

most 1.

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AVL Trees (cont…)

� The balance factor of a node is the height of its left subtree minus the height of its right subtree (sometimes opposite) and a node with balance factor 1, 0, or −1 is considered balanced.

� A node with any other balance factor is considered unbalanced and requires rebalancing the tree.

� The balance factor is either stored directly at each node or computed from the heights of the subtrees.

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AVL Trees

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1

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1412

5 8

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3

72

1

13

1412

5

64AVL tree

not AVL tree

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Single Right Rotation

k2

k1

Xh+1

Yh

Zhk2

k1

Xh+1 Yh Zh

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Single Left Rotation

k2

Xh

k1

Zh+1

Yh

k1

Zh+1

k2

Xh Yh

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Height of AVL Tree

If N is the number of nodes in an AVL tree, the height of the tree, h(N), is approximately 1.44 log(N+2)-.328.

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Inorder Traversal

+

-

*a

/

dc

-

fe

b

(a – (b * (c / d))) + (e – f)

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Method inorder

public static void inorder(BinaryNode t)

{ if ( t!=null )

{ inorder(t.left);

System.out.println(t.element);

inorder(t.right);

}

}

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Preorder Traversal

+

-

*a

/

dc

-

fe

b

+ – a * b / c d – e f

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Method preorder

public static void preorder(BinaryNode t)

{ if ( t!=null )

{ System.out.println(t.element);

inorder(t.left);

inorder(t.right);

}

}

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Postorder Traversal

+

-

*a

/

dc

-

fe

b

a b c d / * – e f – +

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Method postorder

public static void postorder (BinaryNodet)

{ if ( t!=null )

{ inorder(t.left);

inorder(t.right);

System.out.println(t.element);

}

}

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B tree� N-ary tree

� Increase the breadth of trees to decrease the height

� Used for indexing of large amount of data (stored in disk)

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B tree (cont…)

� Unlike a binary-tree, each node of a b-tree may have a variable number of keys and children.

� The keys are stored in non-decreasing order.

� Each key has an associated child that is the root of a subtree containing all nodes with keys less than or equal to the key but greater than the preceeding key.

� A node also has an additional rightmost child that is the root for a subtree containing all keys greater than any keys in the node.

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Example

12 52 78

83 91

60 6919 26 37 46

4 8

012

70717677

7980818283

858690

9395979899

5456575960

61626667

13141719

20212226

27283135

384445

4950

567

891112

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B tree (cont…)

� A b-tree has a minumum number of allowable children for each node known as the minimization factor.

� If t is this minimization factor, every node must have at least t - 1 keys.

� Under certain circumstances, the root node is allowed to violate this property by having fewer than t - 1 keys.

� Every node may have at most 2t - 1 keys or, equivalently, 2t children.

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Height of B-Tree

� For n greater than or equal to one, the

height of an n-key b-tree T of height h with

a minimum degree t greater than or equal

to 2,

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Properties of B Trees

For an M-ary B tree:

� The root has up to M children.

� Non-leaf nodes store up to M-1 keys, and

have between M/2 and M children, except

the root.

� All data items are stored at leaves.

� All leaves have to same depth, and store

between L/2 and L data items.

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Search

12 52 78

83 91

60 6919 26 37 46

4 8

012

70717677

7980818283

858690

9395979899

5456575960

61626667

13141719

20212226

27283135

384445

4950

567

891112

Search for 66

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InsertInsert 55

12 52 78

83 91

60 6919 26 37 46

4 8

012

70717677

7980818283

858690

9395979899

5456575960

61626667

13141719

20212226

27283135

384445

4950

567

891112

Split leave

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Insert

12 52 78

83 91

60 6919 26 37 46

4 8

012

70717677

7980818283

858690

9395979899

5456575960

61626667

13141719

20212226

2728313536

384445

4950

567

891112

Insert 32Split leave

Insert key 31

Insert key 31

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B+ tree

� B+ tree or B plus tree is a type of tree which represents sorted data in a way that allows for efficient insertion, retrieval and removal of records, each of which is identified by a key.

� It is a dynamic, multilevel index, with maximum and minimum bounds on the number of keys in each index segment (usually called a "block" or "node").

� In a B+ tree, in contrast to a B-tree, all records are stored at the leaf level of the tree; only keys are stored in interior nodes.

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B+ tree (cont…)

� The primary value of a B+ tree is in storing data for efficient retrieval in a block-oriented storage context — in particular, file systems.

� This is primarily because unlike binary search trees, B+ trees have very high fanout (typically on the order of 100 or more), which reduces the number of I/O operations required to find an element in the tree.

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A simple B+ tree example linking the keys 1–7 to data values d1-d7. The linked list (red) allows rapid in-order traversal.

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References

� Abdisalam Issa-Salwe, Taibah University, Madinah, Saudi Arabia.

�