Infix to Postfix Converter

This tool implements the shunting-yard algorithm to convert an infix expression (the normal "A + B" form with implied operator precedence) into postfix notation, where operators follow their operands and no parentheses or precedence rules are needed to evaluate it.

Worked example

Enter A+B*C. The algorithm scans left to right, maintaining an output list and an operator stack:

• Read A → output: [A]
• Read + → stack is empty, push it → stack: [+]
• Read B → output: [A, B]
• Read * → compare precedence with the top of stack (+ has precedence 1, * has precedence 2); since * outranks +, don't pop, just push → stack: [+, *]
• Read C → output: [A, B, C]
• End of input: pop remaining operators to output in stack order → pop * then +

Final postfix result: ABC*+. Reading it back confirms correctness: multiply B and C first (respecting the original * before + precedence), then add the result to A — exactly matching the meaning of the original A+B*C.

Why the algorithm pops on equal precedence for left-associative operators

For same-precedence operators like + and - (both left-associative), the converter pops the existing operator off the stack before pushing the new one, which preserves left-to-right evaluation order. The exponentiation operator ^ is right-associative, so the code skips that pop when the incoming operator is ^, which is why A^B^C converts to AB C^^ evaluated as A^(B^C) rather than (A^B)^C.

Validation before conversion

Before running the algorithm, the input is checked for balanced parentheses, no consecutive operators (like "A++B"), and no leading/trailing operators (except a parenthesis) — catching malformed expressions with a specific error message rather than producing garbage output.

Where this is used

Postfix (also called Reverse Polish Notation) is how calculators and interpreters historically evaluated expressions using a simple stack machine, without needing to track parentheses or precedence at evaluation time — the hard part is done once, during this conversion step.