02. binary tree

8/8/2019 02. Binary Tree

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(C) GOYANI MAHESH(C) GOYANI MAHESH 11

DATASTRUCTURES

MAHESH GOYANI

MAHATMAGANDHI INSTITUE OF TECHNICALEDUCATION & RESEARCH CENTER

[email protected]


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BINARY TREE


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BINARY TREE

A binary tree is a finite set of elements that is either empty or ispartitioned into three disjoint subsets.

The first subset contains a single element called the root of the tree.

The other two subsets are themselves binary trees called the left and right

sub trees.

A Binary Tree With n nodes, n>0 has exactly n-1 edges.

A Binary tree of depth d, d>0 has at least d and at most2d 1 nodes

If a binary tree contains n node at level l, than it contains at most 2n

nodes at level l+1


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LEVEL OF BINARY TREE

Root has level 0,

Level of any other node is one more than the level of its parent

A

B

D

H

C

E F

G I

1

0

1

2 2 2

3 3 3

Level 0

Level 1

Level 2

Level 3


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Array Representation

Suppose a binary tree T of depth d then at most 2d-1 nodes can be there in T.So the array of size SIZE is used to represent the binary tree.

For depth 3,

SIZE = (23-1)= 7, so A[7] is enough to store the tree

A

B

D

C

E G

0

1 2

3 4 5 6

A B C D E GA[ ]

[0] [1] [2] [3] [4] [5] [6]


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The father of a node having index n can be obtained by (n-1)/2. For example to find thefather of D, Index of D is 3. Then the father node can be obtained

= (3-1) / 2

= 2 / 2

= 1, i.e, A[1] is father of D, that is B.

The left child of a node having index n can be obtained by (2n + 1). For example to find theleft child of C, index of C = 2

= (2*2 + 1)

=5, i.e A[5] is Left child of C, that is NULL

The Right child of a node having index n can be obtained by (2n + 2). For example to findthe Right child of B, index of B = 1

= (2*1 + 2)

=4, i.e A[4] is Left child of C, that is E

Left and Right brother (Siblings) of node having index n are at location n-1 and n+1respectively.


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Linked List Representation

LChild DATA RChild

A

B NULL C

NULL F NULLNULL E NULLNULL D NULL


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Struct node

{

};

ROOT

LChild RChild

int ROOT;

Struct node *LChild;

Struct node *RChild;

Node Creation

ROOTLChild RChild


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If a binary tree has only LEFT sub tree, then it is called Left Skewed Binary Tree

If a binary tree has only RIGHT sub tree, then it is called Right Skewed Binary Tree

X

Z

Y

A

C

BLEFTSKEWED

RIGHTSKEWED

A

B

D

H

C

E F

G I

BINARYTREE


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A

B

D

H

C

E F

G I

Left sub tree

root

Right sub tree

BINARY TREE


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BINARY TREE

Recursive Definition

B

D

H

C

E F

G ILeft sub tree

root

Right sub tree

A


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BINARY TREE


A

B

D

H

C

E F

G I

Left sub tree

root


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BINARY TREE


A

B

D

H

C

E F

G I

root

Right sub treeLeft sub tree


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NOT A TREE

A

B

D

H

C

E F

G I


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A

B

D

H

C

E F

G I

NOT A TREE


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NOT A TREE

A

B

D

H

C

E F

G I


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OPERATION ON BINARY TREE

There are a number of operations that can be defined for a binary tree.

If p is pointing to a node in an existing tree then

left(p) returns pointer to the left subtree

right(p) returns pointer to right subtree

parent(p) returns the father of p

brother(p) returns brother of p.

info(p) returns content of the node.

In order to construct a binary tree, the following can be useful:

setLeft(p,x) creates the left child node of p. The child node contains the info x.

setRight(p,x) creates the right child node of p. The child node contains the info x.


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APPLICATION OF BINARY TREE

A binary tree is a useful data structure when two-way decisions must be

made at each point in a process.

For example, suppose we wanted to find all duplicates in a list ofnumbers:

14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

One way of finding duplicates is to compare each number with all those

that precede it.

14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5


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The first number in the list is placed in a node that is designated as the root ofa binary tree.

Initially, both left and right sub trees of the root are empty.

We take the next number and compare it with the number placed in the root.

If it is the same then we have a duplicate.

Otherwise, we create a new tree node and put the new number in it.

The new node is made the left child of the root node if the second number isless than the one in the root.

The new node is made the right child if the number is greater than the one inthe root.

CREATION OF BINARY TREE


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If the list of numbers is large and is growing, this procedure involves a largenumber of comparisons.

A linked list could handle the growth but the comparisons would still be large.

The number of comparisons can be drastically reduced by using a binary tree.

The tree grows dynamically like the linked list.

SEARCHING FOR DUPLICATES


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14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 514, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

14



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15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 515, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

1415



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15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 515, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

14

15


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4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 54, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

14

15

4


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4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 54, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

14

154


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9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 59, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

14

154

9


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9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 59, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

14

154

9


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7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 57, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

14

154

9

7


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7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 57, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

14

154

9

7


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18, 3, 5, 16, 4, 20, 17, 9, 14, 518, 3, 5, 16, 4, 20, 17, 9, 14, 5

14

154

9

7

18


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18, 3, 5, 16, 4, 20, 17, 9, 14, 518, 3, 5, 16, 4, 20, 17, 9, 14, 5

14

154

9

7

18


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3, 5, 16, 4, 20, 17, 9, 14, 53, 5, 16, 4, 20, 17, 9, 14, 5

14

154

9

7

18

3


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3, 5, 16, 4, 20, 17, 9, 14, 53, 5, 16, 4, 20, 17, 9, 14, 5

14

154

9

7

183


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5, 16, 4, 20, 17, 9, 14, 55, 16, 4, 20, 17, 9, 14, 5

14

154

9

7

183

5


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5, 16, 4, 20, 17, 9, 14, 55, 16, 4, 20, 17, 9, 14, 5

14

154

9

7

183

5


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16, 4, 20, 17, 9, 14, 516, 4, 20, 17, 9, 14, 5

14

154

9

7

183

5

16


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16, 4, 20, 17, 9, 14, 516, 4, 20, 17, 9, 14, 5

14

154

9

7

183

5

16


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4, 20, 17, 9, 14, 54, 20, 17, 9, 14, 5

14

154

9

7

183

5

16

4


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20, 17, 9, 14, 520, 17, 9, 14, 5

14

154

9

7

183

5

16

20


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20, 17, 9, 14, 520, 17, 9, 14, 5

14

154

9

7

183

5

16 20


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TRACE OF INSERT

17, 9, 14, 517, 9, 14, 5

14

154

9

7

183

5

16 20

17pq


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17, 9, 14, 517, 9, 14, 5

14

154

9

7

183

5

16 20

17p

q

TRACE OF INSERT


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TRACE OF INSERT

17, 9, 14, 517, 9, 14, 5

14

154

9

7

183

5

16 20

17

p

q


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TRACE OF INSERT

17, 9, 14, 517, 9, 14, 5

14

154

9

7

183

5

16 20

17

p

q


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TRACE OF INSERT

17, 9, 14, 517, 9, 14, 5

14

154

9

7

183

5

16 20

17

pq


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TRACE OF INSERT

17, 9, 14, 517, 9, 14, 5

14

154

9

7

183

5

16 20

17

p

q


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TRACE OF INSERT

17, 9, 14, 517, 9, 14, 5

14

154

9

7

183

5

16 20

17

pq


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TRACE OF INSERT

17, 9, 14, 517, 9, 14, 5

14

154

9

7

183

5

16 20

17

p

q


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TRACE OF INSERT

17, 9, 14, 517, 9, 14, 5

14

154

9

7

183

5

16 20

17

p

p->setRight( node );

node

TREE TRAVSERSAL


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TREE TRAVSERSAL14

154

(4,14,15), (4,15,14)

(14,4,15), (14,15,4)

(15,4,14), (15,14,4)

Three common ways:node

Preorder: (N,L,R)

Inorder: (L,N,R)

Postorder: (L,R,N)left

subtreeright

subtreeL

N

R


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3

1

2

5

6

7 9

8

4

INORDER


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INORDER

Inorder: 3 4 5 7 9 14 15 16 17 18 20Inorder: 3 4 5 7 9 14 15 16 17 18 20

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183

5

16 20

17


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Make Money Fast!

1. Motivations References2. Methods

2.1 StockFraud

2.2 PonziScheme

1.1 Greed 1.2 Avidity2.3 BankRobbery

1

2

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5

46 7 8

9

PREORDER


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Preorder: 14 4 3 9 7 5 15 18 16 17 20Preorder: 14 4 3 9 7 5 15 18 16 17 20

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183

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16 20

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PREORDER


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cs16/

homeworks/ todo.txt1K

programs/

DDR.java10K

Stocks.java25K

h1c.doc3K

h1nc.doc2K

Robot.java20K

9

3

1

7

2 4 5 6

8

POSTORDER


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POSTORDER

Postorder: 3 5 7 9 4 17 16 20 18 15 14Postorder: 3 5 7 9 4 17 16 20 18 15 14

14

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9

7

183

5

16 20

17

COMPLEXITY


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COMPLEXITY

Average Case : O (log2n)

Best Case : O (log2n)

Worst Case : O (log2n)

BINARY SEARCH TREE


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BINARY SEARCH TREE

Binary Search Tree (BST) is a binary tree, which is either empty or satisfiesfollowing properties :

Every node has a value and no two nodes have the same property

If there exists a LEFT CHILD then its value is less than the value ofROOT.

If there exists a RIGHT CHILD then its value is greater than the valueofROOT.

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02. binary tree

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