Infix to Postfix Converter Online – Step by Step Evaluation

How to Convert Infix to Postfix Expression Using Stack Algorithm

An infix to postfix converter transforms mathematical and logical expressions written in infix notation (operators between operands, e.g., A + B) into postfix notation (Reverse Polish Notation or RPN, e.g., A B +). This conversion enables simple stack-based evaluation without parentheses, making it ideal for calculators, compilers, interpreters, and expression evaluators used in data structures coursework and real-world software engineering.

Our free infix to postfix converter online tool above instantly converts any valid infix expression to its postfix equivalent and displays a complete step-by-step trace showing the output queue and operator stack at every stage. Whether you’re studying for a data structures exam, debugging a compiler, or simply curious about how the Shunting Yard algorithm works, this tool has you covered.

Infix to Postfix Conversion Formula and Rules

Core Variables and Definitions

• Expression tokens: A sequence T = t₁, t₂, …, tₙ where each token tᵢ is an operand, operator, left parenthesis ( or right parenthesis ).
• S: A stack used for operators and parentheses.
• O: An output list (the postfix sequence).
• prec(op): The precedence value of operator op (higher number = higher precedence).
• assoc(op): The associativity of operator op — either left or right.

Conversion Rules (Shunting Yard Algorithm)

Given tokens t₁ through tₙ, process each token using the following rules:

1. If tᵢ is an operand: append it to the output O.
2. If tᵢ is a left parenthesis (: push it onto the stack S.
3. If tᵢ is a right parenthesis ): pop operators from S and append to O until ( is found, then discard the (.
4. If tᵢ is an operator o₁: while S is not empty and the top of S is an operator o₂ and either (assoc(o₁) = left AND prec(o₁) ≤ prec(o₂)) or (assoc(o₁) = right AND prec(o₁) < prec(o₂)), pop o₂ from S and append to O. Then push o₁ onto S.
5. After all tokens are processed: pop all remaining operators from S and append them to O.

Operator Precedence and Associativity Table

Step-by-Step Infix to Postfix Conversion Examples

Symbolic Example: (A+B)*C – D/E

Let’s trace through the conversion of (A+B)*C - D/E step by step:

1. ( → Push to stack. Stack: ( | Output: ∅
2. A → Operand, output it. Stack: ( | Output: A
3. + → Push to stack. Stack: ( + | Output: A
4. B → Operand, output it. Stack: ( + | Output: A B
5. ) → Pop until (. Pop + to output. Stack: ∅ | Output: A B +
6. * → Push to stack. Stack: * | Output: A B +
7. C → Operand, output it. Stack: * | Output: A B + C
8. - → prec(-) ≤ prec(*), pop * to output, push -. Stack: - | Output: A B + C *
9. D → Operand, output it. Stack: - | Output: A B + C * D
10. / → prec(/) > prec(-), push /. Stack: - / | Output: A B + C * D
11. E → Operand, output it. Stack: - / | Output: A B + C * D E
12. End → Pop all: / then -. Final Output: A B + C * D E / -

Numeric Example: (3+4)*5 – 6/2

Following the same algorithm, (3+4)*5 - 6/2 converts to postfix: 3 4 + 5 * 6 2 / -

Evaluating: 3+4=7, 7*5=35, 6/2=3, 35-3 = 32.

How to Evaluate Postfix Expressions Using a Stack

Once you have a postfix expression, evaluating it is straightforward with a stack:

1. Initialize an empty evaluation stack E.
2. For each token t in the postfix sequence: ◦ If t is a number, push it onto E. ◦ If t is an operator, pop two values (b then a), compute a op b, push the result.
3. The final value in E is the result.

Practical Applications of Infix to Postfix Conversion

• Compilers and interpreters: Convert infix expressions into postfix or ASTs for efficient code generation.
• Embedded systems: Postfix evaluation avoids recursion, ideal for constrained devices.
• Scientific and RPN calculators: Many HP calculators use RPN internally for fast evaluation.
• Spreadsheets and databases: Expression evaluators use conversion to avoid repeated parsing.
• Math and symbolic manipulation systems: Convert between representations for algebraic simplification.

Common Pitfalls in Infix to Postfix Conversion

• Treating - as always binary leads to errors with expressions like -3*2.
• Using an incorrect precedence table (e.g., swapping * and +) produces wrong results.
• Forgetting to flush the stack at the end yields incomplete postfix output.
• Inconsistent division (integer vs. floating-point) during evaluation.

FAQ

What does “Infix to Postfix Converter” mean?

An infix to postfix converter is a tool or algorithm that transforms expressions written in standard infix notation (operators between operands) into postfix notation (operators after operands, also called Reverse Polish Notation). This conversion simplifies evaluation because it removes the need for parentheses and respects operator precedence explicitly via ordering.

What is the time complexity of infix to postfix conversion?

The Shunting Yard algorithm runs in O(n) time where n is the number of tokens, because each token is pushed and popped at most a constant number of times.

Can this tool handle multi-digit numbers?

This tool is designed for single-character operands (A-Z, a-z, 0-9) which is the standard format used in academic data structures courses. For multi-digit numeric evaluation, you would need to tokenize the input differently.