Expression Trees and Huffman Encoding, Slides of Data Structures and Algorithms

Download Expression Trees and Huffman Encoding and more Slides Data Structures and Algorithms in PDF only on Docsity!

1

Expression Tree

 The inner nodes contain operators while leaf

nodes contain operands.

a

c

b

g

d

e

f

2

Expression Tree

 The tree is binary because the operators are

binary.

a

c

b

g

d

e

f

4

Expression Tree

 Inorder traversal yields: a+bc+de+f*g

a

c

b

g

d

e

f

5

Enforcing Parenthesis

void inorder(TreeNode* treeNode)

if( treeNode != NULL ){

cout << "(";

inorder(treeNode->getLeft());

cout << ")";

cout << *(treeNode->getInfo());

cout << "(";

inorder(treeNode->getRight());

cout << ")";

7

Expression Tree

 Postorder traversal: a b c * + d e * f + g * +

which is the postfix form.

a

c

b

g

d

e

f

8

Constructing Expression Tree

 Algorithm to convert postfix expression into an

expression tree.

 We already have an expression to convert an

infix expression to postfix.

 Read a symbol from the postfix expression.

 If symbol is an operand, put it in a one node tree

and push it on a stack.

 If symbol is an operator, pop two trees from the

stack, form a new tree with operator as the root

and T 1 and T 2 as left and right subtrees and

push this tree on the stack.

10

Constructing Expression Tree

 a b + c d e + * *

a b

Stack is growing left to right

If symbol is an operand, put it in a one node tree and push it on a stack.

top

11

Constructing Expression Tree

 a b + c d e + * *

a b

Stack is growing left to right

If symbol is an operator, pop two trees from the stack, form a new tree with operator as the root and T 1 and T 2 as left and right subtrees and push this tree on the stack.

13

Constructing Expression Tree

 a b + c d e + * *

a b

• c

d e

14

Constructing Expression Tree

 a b + c d e + * *

a b

c

d e

16

Other Uses of Binary Trees

Huffman Encoding

17

Huffman Encoding

 Data compression plays a significant role in

computer networks.

 To transmit data to its destination faster, it is

necessary to either increase the data rate of the

transmission media or to simply send less data.

 Improvements with regard to the transmission

media has led to increase in the rate.

 The other options is to send less data by means

of data compression.

 Compression methods are used for text, images,

voice and other types of data (space probes).

19

Huffman Encoding

 To understand Huffman encoding, it is

best to use a simple example.

 Encoding the 32-character phrase:

" traversing threaded binary trees ",

 If we send the phrase as a message in a

network using standard 8-bit ASCII codes,

we would have to send 8*32= 256 bits.

 Using the Huffman algorithm, we can send

the message with only 116 bits.

20

Huffman Encoding

 List all the letters used, including the "space"

character, along with the frequency with which

they occur in the message.

 Consider each of these (character,frequency)

pairs to be nodes; they are actually leaf nodes,

as we will see.

 Pick the two nodes with the lowest frequency,

and if there is a tie, pick randomly amongst

those with equal frequencies.