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Infix to Postfix Cheat Sheet

Everything you need for an exam answer on one page: operator precedence, the seven shunting-yard rules, worked examples and the mistakes that cost marks.

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1 · Operator precedence

Higher binds tighter. Everything on one row has equal precedence.

Level	Operators	Associativity
3 (highest)	`^`	Right to left
2	`*` `/` `%`	Left to right
1 (lowest)	`+` `-`	Left to right

Brackets are not an operator — they override precedence entirely, and never appear in the postfix output.

2 · The algorithm, in seven rules

Operand — send it straight to the output.
Left bracket ( — push it on the stack.
Right bracket ) — pop to the output until the matching ( appears; discard both brackets.
Operator, stack top has higher precedence — pop the top to the output, then test again.
Operator, equal precedence and left-associative — pop, then test again.
Operator, equal precedence and right-associative (^) — do not pop; push the new operator.
End of input — pop everything left on the stack to the output.

The stack only ever holds operators and left brackets. Operands never touch it.

3 · Worked examples

Infix	Postfix	What it tests
`A + B * C`	`A B C * +`	Precedence: `*` before `+`
`(A + B) * C`	`A B + C *`	Brackets beat precedence
`A + B - C`	`A B + C -`	Equal precedence, left to right
`A ^ B ^ C`	`A B C ^ ^`	Right-associativity of `^`
`A * B + C / D`	`A B * C D / +`	Two equal-precedence pairs

4 · Mistakes that cost marks

Treating ^ as left-associative. A ^ B ^ C is A B C ^ ^, not A B ^ C ^.
Leaving brackets in the output. Postfix never contains a bracket — that is the whole point of it.
Putting % with ^. Modulo sits with * and / at level 2.
Forgetting the final pop. Operators left on the stack at the end still have to reach the output, deepest last.
Popping the ( to the output. On a ) the matching ( is discarded, not emitted.

5 · Reading it back

To evaluate postfix, scan left to right with a second stack: push operands, and on an operator pop two values — second popped, operator, first popped — then push the result. Order matters for - and /.

Prefix is the mirror image: reverse the infix expression, swap every bracket, run the same algorithm, then reverse the output.

infixtopostfixconverter.com — check any expression against the converter or drill yourself with the practice questions.