51658218 Non Linear Data Structures

8/2/2019 51658218 Non Linear Data Structures

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Non-linear data structures

TREES


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TREES

Non-linear data structure

Use to represent hierarchicalrelationship among existing data

items.


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Trees TerminologyNode Each data item in a tree.

Root no parent. The topmostnode (A)

child nodes Each node in a tree

has zero or more nodes B,C,D

A

B C

E F G H

D

Root node

Leaf nodes

Interior nodes Height

edge


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Terminology parent node A node that has a child is

called the child's parent node

An internal node or inner node

is any node of a tree that has childnodes.

Depth of tree distance from root toleaf node. In given tree it is 3.

The depth ofa node n is the length

of the path from the root to the node


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Degree of node :

It is the number of subtrees of nodein given tree. Degree of node B is

one, A & C is 3.Degree of tree

It is the maximum degree of nodes in

given tree.

In given tree degree of node b is oneand A&C is 3 so max degree is 3


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Non terminal nodes Any node whosedegree is not zero

Leaf or Terminal node no child

Siblings

The children nodes of given parent node.They are also called brothers


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Level

The entire tree is leveled in suchway that root node is at level 0, its

immediate children are at level 1and their immediate children atlevel two and so on.

i.e. if node is at level n, then itschildren will be at level n+1.


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Edge

Line drawn from one node toanother node

Path

It is the sequence of consecutiveedges from source to thedestination node.

The path between A to F is (A,C)


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Strictly binary trees

A binary tree is Strictly binary treeif and only if :-

Each node has exactly two child

nodes or no nodes


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full binary

A binary tree is afull binary tree ifand only if

Each non leafnode has exactlytwo child nodes

All leaf nodes areat the same level


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Complete binary treeA binary tree is a complete binary tree (of

height h , we take root node as 0 ) if and only if:-

Level 0 to h-1 represent a full binary tree ofheight h-1

One or more nodes in level h-1 may have 0,or 1 child nodes If j ,k are nodes in level h-1, then j has morechild nodes than k if and only if j is to the left ofk


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Extended binary tree

A binary tree T is said to be 2-tree orextended binary tree if each nodeN has either 0 or 2 children.

In such case, node with 2 children are

called internal nodes and the nodeswith 0 children are called externalnodes.

Binary tree

extendedbinarytree


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Binary tree representation

Root node index starts with 0

A

B C

D E F G

0

21

3 4 5 6

0 A

1 B

2 C3 D

4 E

5 F

6 G


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LINK Representation

Each node consist ofData

Left child

Right child

Lchild Data Rchild


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LINK Representation

a

cb

d

f

e

g

hleftChildelement

rightChild

root


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Maximum Number OfNodes

Maximum number ofnodes

= 1 + 2 + 4 + 8 + +2h-1

A full binary tree of a given height hhas 2h 1 nodes.


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It is the way in which each node inthe tree is visited exactly once insystematic manner.

Used to represent arithmeticexpressions.

Preorder

In order

Post order

Binary Tree TraversalMethods


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Tree traversals

The order in which the nodes are visited duringa tree traversal,

preorderpreorder inorderinorder postorderpostorder

In preorder, the root is visited first

In inorder, the root is visited in the middle

In postorder, the root is visited last


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Pre-order traversalStart at the root node

Traverse the left subtree

Traverse the right subtree

The nodes of this treewould be visited in the

order :

D B A C F E G


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Post-order traversalTraverse the left subtree

Traverse the right subtree

Visit the root node

A C B E G F D


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Post-order


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Conversion of expressioninto binary treeA+B


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1.A+B*C2. A/B-C


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1.(A-B)+C2. A-(B+C)


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(A+B)*(C-D)

raw ree:


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A/B+C/D

(A+B+C)*(D+E+F)

raw ree:-Write preorder, postorder,

inorder


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Inorder: EACKFHDBGpreorder: FAEKCDHGB

FA D

E K H GC B


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Binary Search Tree

The value of the key in the leftchild or left subtree is less thanvalue of the root

The value of the key in rightsubtree is more than or equal tothe value of the root.

All subtree of the left and rightchildren observe these two rules.


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binary search tree (BST) is anode based binary tree data

structure

sometimes also be called orderedor sorted binary tree

Searching


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SearchingCan be recursive or iterative process.

begin by examining the root node. If the tree is null, thevalue we are searching for does not exist in the tree.

Otherwise, if the value equals the root, the search issuccessful.

If the value is less than the root, search the left subtree.

Similarly, if it is greater than the root, search the rightsubtree.

This process is repeated until the value is found or theindicated subtree is null.

If the searched value is not found before a null subtreeis reached, then the item must not be present in thetree.


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insertion

Check the root and recursivelyinsert the new node to the leftsubtree if the new value is less

than the root, or the right subtree ifthe new value is greater than theroot.


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Inserts the node pointed to by"newNode" into the subtree rooted

at "treeNode" */void InsertNode(Node* &treeNode,Node *newNode)

{

if (treeNode == NULL)treeNode = newNode;else if (newNode->key < treeNode->key) InsertNode(treeNode->left,

newNode);else InsertNode(treeNode->right,newNode);}


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Deletion

Deleting a leaf (node with nochildren): Deleting a leaf is easy,as we can simply remove it fromthe tree.

Deleting a node with one child:Remove the node and replace itwith its child.

Deleting a node with twochildren: Call the node to bedeleted N. Do not delete N.

Instead, choose either its in-order


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B-trees

B-trees are balanced sort trees thatare optimized for situations whenpart or all of the tree must bemaintained in secondary storage

such as a magnetic disk.

The B-tree algorithm minimizes thenumber of times a medium must

be accessed to locate a desiredrecord, thereby speeding up theprocess.


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B-tree

A binary-tree, each node of a b-treemay have a variable number ofkeys and children.

The keys are stored in non-decreasing order


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Each key has an associated child thatis the root of a subtree containing allnodes with keys less than or equal tothe key but greater than thepreceeding key.

A node also has an additionalrightmost child that is the root for a

subtree containing all keys greaterthan any keys in the node.


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Searching techniques


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Searching techniques

Process of finding elements within listof elements.

Two categories:-

Linear search No prerequisite

Binary search

List must be sorted.


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Linear search

Access an element of an array oneby one sequentially.

Search unsuccessful if element is

not found.

Average case, we may have toscan half of the list.


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Binary search

Searches item in min comparisons,This tech,

First find middle element (mid=first

+ last)/2)Compare mid element with an item

There are three cases,

If mid element=search element thensearch is successful.

If it is less than desired item then search

only first half array, beg=mid-1


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9,12,24,30,36,45,70

Item=45

Beg=0 and last=6

Mid=int(0+6)/2=3

A[3]=30

45>30 then

Beg=mid+1=3+1

=4


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New mid is,

Int(4+6)=5

A[5]=45

Search is successful


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12,14,25,32,35,40,45,48,50Item search=25

51658218 Non Linear Data Structures

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