Mastering Prefix, Infix, and Postfix Conversion Using Stack and Queue

Mathematical expressions are a crucial part of programming, and understanding their different forms—Prefix, Infix, and Postfix—is essential for expression evaluation, compiler design, and algorithm development. Stack and Queue play a vital role in efficiently converting between these notations.

In this guide, you’ll learn:
✅ What Prefix, Infix, and Postfix Notations are
✅ How to convert between them using Stack and Queue
✅ Algorithms and Python implementations
✅ Real-world applications and key differences

By the end, you’ll have a solid understanding of expression conversion and how to implement it in programming.

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What Are Prefix, Infix, and Postfix Notations?

1. Infix Notation (Human-Readable Form)

In Infix notation, operators are placed between operands. This is the standard notation we use in mathematics.

Example:

(A + B) * C

✅ Easy to read for humans
❌ Requires parentheses to ensure correct precedence

2. Prefix Notation (Polish Notation)

In Prefix notation, also called Polish notation, operators appear before operands.

Example:

* + A B C

✅ No need for parentheses
❌ Harder to read for humans

3. Postfix Notation (Reverse Polish Notation – RPN)

In Postfix notation, operators appear after operands.

Example:

A B + C *

✅ No need for parentheses
✅ Easier for computers to evaluate using Stack
❌ Harder for humans to interpret

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Why Use Stack and Queue for Conversion?

Since Stack and Queue are linear data structures that follow LIFO (Last In, First Out) and FIFO (First In, First Out) principles, they are ideal for handling operator precedence and association when converting expressions.

Use Cases in Expression Conversion:

🔹 Stack: Used to hold operators and ensure correct precedence.
🔹 Queue: Used in some approaches for handling postfix expressions efficiently.

Algorithm for Converting Infix to Postfix (Stack Method)

1. Initialize an empty stack for operators and an empty output string.
2. Traverse the infix expression from left to right.
3. If the current character is an operand, append it to the output.
4. If the current character is an operator:
   • Pop from the stack while operators have higher or equal precedence.
   • Push the current operator onto the stack.
5. If the character is ‘(‘, push it onto the stack.
6. If the character is ‘)’, pop and append until ‘(‘ is encountered.
7. Pop and append remaining operators from the stack.

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Python Implementation: Infix to Postfix

def precedence(op):
    if op in ('+', '-'):
        return 1
    if op in ('*', '/'):
        return 2
    return 0

def infix_to_postfix(expression):
    stack = []
    output = ''
    
    for char in expression:
        if char.isalnum():
            output += char
        elif char == '(':
            stack.append(char)
        elif char == ')':
            while stack and stack[-1] != '(':
                output += stack.pop()
            stack.pop()
        else:
            while stack and precedence(stack[-1]) >= precedence(char):
                output += stack.pop()
            stack.append(char)
    
    while stack:
        output += stack.pop()
    
    return output

# Example Usage
expr = "A+B*(C-D)"
print(infix_to_postfix(expr))  # Output: ABCD-*+

Algorithm for Converting Infix to Prefix

1. Reverse the infix expression.
2. Convert reversed infix to postfix.
3. Reverse the result to get the prefix expression.

Python Implementation: Infix to Prefix

def infix_to_prefix(expression):
    expression = expression[::-1]
    expression = expression.replace('(', 'temp').replace(')', '(').replace('temp', ')')
    postfix = infix_to_postfix(expression)
    return postfix[::-1]

# Example Usage
expr = "(A+B)*C"
print(infix_to_prefix(expr))  # Output: *+ABC

Algorithm for Converting Postfix to Infix

1. Initialize an empty stack.
2. Traverse the postfix expression:
   • If operand, push onto stack.
   • If operator, pop two elements, form a new expression, push back.
3. Final expression on stack is the infix expression.

Python Implementation: Postfix to Infix

def postfix_to_infix(expression):
    stack = []
    
    for char in expression:
        if char.isalnum():
            stack.append(char)
        else:
            op1 = stack.pop()
            op2 = stack.pop()
            stack.append(f"({op2}{char}{op1})")
    
    return stack[-1]

# Example Usage
expr = "AB+C*"
print(postfix_to_infix(expr))  # Output: ((A+B)*C)

Key Differences: Prefix, Infix, Postfix

Feature	Infix	Prefix	Postfix
Operator Position	Between operands	Before operands	After operands
Readability	Easy	Harder	Harder
Parentheses Required?	Yes	No	No
Used By	Humans	Compilers	Stack-based Evaluation

Real-World Applications of Expression Conversion

✅ Compilers – Convert infix expressions to postfix for evaluation.
✅ Calculators – Evaluate expressions using Stack and Queue.
✅ Reverse Polish Notation (RPN) Calculators – Used in HP calculators.
✅ Mathematical Expression Solvers – Used in AI-based calculation apps.

FAQs

Conclusion

Understanding Prefix, Infix, and Postfix Conversion using Stack and Queue is essential for programming, compilers, and mathematical computations.

🚀 Key Takeaways:
✅ Use Stacks for conversion efficiently.
✅ Infix is easy for humans, but Postfix is better for machines.
✅ Compilers, calculators, and AI tools rely on Postfix evaluation.

Master this concept to excel in data structures, algorithms, and coding interviews!