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Lecture 6

C S 1 1 3 2 D A T A S T R U C T U R E S A N D S O F T W A R ED E S I G N

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2

Introduction to Graphs

A graph is a finite set of nodes with edges between nodes

Formally, a graph G is a structure (V,E) consisting of

a finite set V called the set of nodes, and a set E that is a subset of VxV. That is, E is a set of pairs of the form (x,y) where x

and y are nodes in V

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3

Examples of GraphsV={1,2,3,4,5}E={(1,2), (2,3), (1,4), (4,1), (3,3), (5,4)}

12

5

3

4

When (x,y) is an edge,

we say that x is adjacent

to y.

1 is adjacent to 2.

2 is not adjacent to 1.

4 is not adjacent to 3.

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4

Trees

A tree is a connected acyclic undirected graph. The following are threetrees:

12

5

311

12

10

98

7

64

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Make binary tree by sorted

ArrayArray A[15] = { 1, 4,8,11,14,19,23,29,40,89,91,105,110,124,200}

8

29

11

4 19

14 23

105

124

91 2001

89

40 110

A[15] :

The First is A[0]

And last is A[14]

So (0+14)/2 is

A[7] which is equal

To 29

And Now we have two array one of the is bigger than 29 that is in the right side

And the other one has 7 int that all of the are smaller than 29

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Illustration of Insert

6

15

8

2

3 7

11

10

14

12

20

27

22 30

Before inserting 25

6

158

2

3 7

11

10

14

12

20

27

22 30

25

After inserting 25

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10 min for Array To Binary

Search Tree function

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De-allocating binary treesThree things to do

Free current node

Recursively free left subtree

Recursively free right subtree

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C Binary Tree node

struct btnode

{int data;struct btnode *left_p;struct btnode *right_p;

} ;

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Creating a treestruct btnode *root;

struct btnode *mynode =

(struct btnode *) malloc (sizeof(struct btnode));

root = mynode;

// use root to access the tree, like head for a linked list

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Creating nodesstruct node * NewNode(int data)

{struct node *mynode = (struct node *) malloc (sizeof(struct node));

mynode->data = data;mynode->left_p = NULL;mynode->right_p = NULL;

return(node);}

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10 min for t_insert and Search

function

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Insert function

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Binary Tree TraversalTraversal is the process of visiting everynode once

Visiting a node entails doing someprocessing at that node, but whendescribing a traversal strategy, we need

not concern ourselves with what thatprocessing is

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Binary Tree Traversal

TechniquesThree recursive techniques for binary tree traversal

In each technique,

the left subtree is traversed recursively,

the right subtree is traversed recursively,

and the root is visited

What distinguishes the techniques from one anotheris the order of those 3 tasks

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Preoder, Inorder, Postorder

In Preorder, the rootis visited before (pre)

the subtrees traversals

In Inorder, the root isvisited in-between left

and right subtree traversal

In Preorder, the root

is visited after (pre)

the subtrees traversals

Preorder Traversal:

1. Visit the root

2. Traverse left subtree

3. Traverse right subtree

Inorder Traversal:


2. Visit the root


Postorder Traversal:



3. Visit the root

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Illustrations for Traversals

Assume: visiting a node

is printing its label

Preorder:

Inorder:

Postorder:

1

3

11

98

4 6

5

7

12

10

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Illustrations for TraversalsAssume: visiting a node

is printing its label

Preorder:

1 3 5 4 6 7 8 9 10 11 12

Inorder:

4 5 6 3 1 8 7 9 11 10 12

Postorder:

4 6 5 3 8 11 12 10 9 7 1

1

3

11

98

4 6

5

7

12

10

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Illustrations for Traversals (Contd.)

Assume: visiting a node

is printing its data

Preorder: 15 8 2 6 3 7

11 10 12 14 20 27 22 30

Inorder: 2 3 6 7 8 10 11

12 14 15 20 22 27 30

Postorder: 3 7 6 2 10 1412 11 8 22 30 27 20 15

6

15

8

2

3 7

11

10

14

12

20

27

22 30

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Code for the Traversal Techniques

The code for visit

is up to you toprovide, depending

on the application

A typical example

for visit() is to

print out the data

part of its input

node

void inOrder(Tree *tree){

if(tree->isEmpty( )) return;inOrder(tree->getLeftSubtree( ));

visit(tree->getRoot( ));

inOrder(tree->getRightSubtree( ));

}

void preOrder(Tree *tree){

if(tree->isEmpty( )) return;

visit(tree->getRoot( ));preOrder(tree->getLeftSubtree());

preOrder(tree->getRightSubtree());

}

void postOrder(Tree *tree){

if(tree->isEmpty( )) return;

postOrder(tree->getLeftSubtree( ));

postOrder(tree->getRightSubtree( ));

visit(tree->getRoot( ));

}

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Application of TraversalSorting a BST

Observe the output of the inorder traversal ofthe BST example two slides earlier

It is sortedThis is no coincidence

As a general rule, if you output the keys (data)

of the nodes of a BST using inorder traversal,the data comes out sorted in increasing order

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Depth/Height of Full Trees and

Almost Complete TreesThe height (or depth ) h of such trees is O(log n)

Proof: In the case of full trees,

The number of nodes n is: n=1+2+22

+23

++2h

=2h+1

-1 Therefore, 2h+1 = n+1, and thus, h=log(n+1)-1

Hence, h=O(log n)

For almost complete trees, the proof is left as an

exercise.

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Canonical Labeling ofAlmost Complete Binary Trees

Same labeling inherited from full binary trees

Same relationship holding between the labels ofchildren and parents:

Relationships between labels

of children and parent:

2i 2i+1

i

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Trees and arrays

Figure from http://scientopia.org/blogs/goodmath/2008/04/29/implementing-compact-binary-heaps/

Map current node, left child, and right child to array positions

t[i] , t[2i+1] , t[2i+2]

Order in the array

by level in the tree

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Trees and arrays

Figure from http://scientopia.org/blogs/goodmath/2008/04/29/implementing-compact-binary-heaps/

Map current node, left child, and right child to array positions

Order in the array

by level in the tree

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Application of Almost Complete

Binary Trees: HeapsA heap (or min-heap to be precise) is an almost complete binary treewhere

Every node holds a data value (or key)

The key of every node is the keys of the children

Note:

A max-heap has the same definition except that the

Key of every node is >= the keys of the children

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Search function

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Delete a Complete TreeTo delete a tree we must traverse all the nodes ofthe tree and delete them one by one. So whichtraversal we should use Inorder or Preorder or

Postorder. Answer is simple Postorder, becausebefore deleting the parent node we should deleteits children nodes first

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10 min for t_insert function

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