CSC 205Programming II

Postfix Expressions

Recap: Stack

Stack features Orderly linear structure Access from one side only – top item

Stack operations Push – add an item to top Pop – remove the top item Peek – get the content of the top item

Implementations and trade-offs

Array-based Implementation

Instance variables needed Collection of data – Object[] items Two auxiliary variables

• Number of items on stack – int numItems• Stack capacity – int CAPACITY

Selected methods pop popAll

Arithmetic Expressions

Three possible orders Infix: operator in between operands

2 * (3 + 4)• Widely used in normal presentation• Parentheses needed to remove ambiguity

Prefix: operator proceeds operands

* + 3 4 2 Postfix: operator following operands

2 3 4 + *

Evaluating Expressions

Infix 2 * (3 + 4) 2 * (3 + 4) 2 * 7 14

Prefix * + 3 4 2 * + 3 4 2 * 7 2 14

Postfix 2 3 4 + * 2 3 4 + * 2 7 * 14

Evaluating Postfix Expressions Using Stack Assumptions

Correct postfix expression With arithmetic operators +, -, *, and / only

The algorithm Initialize a stack Read the next input while there is one

• If (the next input is a number)• Push it onto the stack

• Else• Pop two numbers off the stack • Carry out the calculation & push the result onto the

stack

Converting Infix to Postfix Notation It’s easier to evaluate postfix expressions

No need for a separate stack for operators: an operator is consumed when encountered

Infix expressions can be converted into equivalent postfix ones then be evaluated

Assumptions Correct infix expression +, -, *, and / only

Sample Expression 1 –fully parenthesized

(a – ((b + (c * d)) / e))

Converting Infix to Postfix Notation – when fully parenthesized

Initialize a stack (for operators and parentheses) Read the next input while there is one

If (the next input is a left parenthesis)• Push it onto the stack

Else if (the next input is a number)• Write it to the output

Else if (the next input is an operator)• Push it onto the stack

Else• Pop and output the top item (an operator)

• Pop and discard the top item (a left parenthesis)

Sample Expression 2 – parenthesized as needed

a – (b + c * d) / e

Converting Infix to Postfix Notation – using precedence rules

Initialize a stack (for operators and parentheses) Read the next input while there is one

If (the next input is a left parenthesis)• Push it onto the stack

Else if (the next input is a number)• Write it to the output

Else if (the next input is an operator)• Pop and print operators off the stack until

• Stack become empty

• The next item is a left parenthesis

• The next item is an operator of lower precedence

• Push the new operator onto stack Else

• Pop and output the top item (an operator)• Pop and discard the top item (a left parenthesis)