📐 Infix · Postfix · Prefix notation, conversion & use

📍 Infix inf

✧ standard form

A + B · (C - D) * E

⟹ operator between operands
human-friendly

Example: (3 + 4) × 5 → 35
needs parentheses & precedence

📎 Postfix post

✧ reverse Polish (RPN)

A B + · C D - E *

⟹ operator after operands
no parentheses needed

Example: 3 4 + 5 × → 35
evaluated left‑to‑right using stack

🔖 Prefix pre

✧ Polish notation

+ A B · * - C D E

⟹ operator before operands
no ambiguity

Example: × + 3 4 5 → 35
operator precedes operands

Same expression (infix): (A + B) * C → A B + C * (postfix) → * + A B C (prefix)

🔄 Detailed conversion: step‑by‑step

We'll convert the infix expression: (A + B) * C - D / E (with precedence: * and / before + and -)

📌 1. Infix → Postfix (Shunting‑yard algorithm)

Rules: Scan left‑to‑right. Operands → output. Operators → stack (higher precedence pops first). Parentheses: ( push, ) pop until (.

Step	Symbol	Stack	Output	Action
1	`(`	`(`		push `(`
2	`A`	`(`	`A`	output operand
3	`+`	`( +`	`A`	push `+`
4	`B`	`( +`	`A B`	output operand
5	`)`		`A B +`	pop until `(` → output `+`
6	`*`	`*`	`A B +`	push `*`
7	`C`	`*`	`A B + C`	output operand
8	`-`	`-`	`A B + C *`	`*` pops, push `-`
9	`D`	`-`	`A B + C * D`	output operand
10	`/`	`- /`	`A B + C * D`	push `/`
11	`E`	`- /`	`A B + C * D E`	output operand
12	end		`A B + C * D E / -`	pop all → `/` then `-`

✅ Postfix result: A B + C * D E / -

📌 2. Infix → Prefix (parse tree / pre‑order)

Method: Build expression tree, then pre‑order traversal (root → left → right).

Tree: - (root) → left * → left + → A B → right C → right / → D E

Step	Traverse	Prefix output	Explanation
1	root `-`	`-`	root operator
2	left `*`	`- *`	left subtree
3	left of `*` → `+`	`- * +`	left of *
4	left of `+` → `A`	`- * + A`	operand
5	right of `+` → `B`	`- * + A B`	operand
6	right of `*` → `C`	`- * + A B C`	operand
7	right of `-` → `/`	`- * + A B C /`	right subtree
8	left of `/` → `D`	`- * + A B C / D`	operand
9	right of `/` → `E`	`- * + A B C / D E`	operand

✅ Prefix result: - * + A B C / D E

📌 3. Postfix ↔ Prefix (stack reversal)

Method: Reverse postfix, scan, and combine using a stack.

Postfix: A B + C * D E / - → Reverse: - / E D * C + B A

Step	Symbol	Stack (prefix)	Action
1	`-`	`-`	push operator
2	`/`	`- /`	push operator
3	`E`	`- / E`	push operand
4	`D`	`- / E D`	push operand
5	`*`	pop D,E → `/ D E`, push `*`	combine
6	`C`	pop `/ D E` and `C` → `* / D E C`	combine
7	`+`	pop `* / D E C` and `B` → `+ B * / D E C`	combine
8	`B`	(already combined)	push operand
9	`A`	pop `+ B * / D E C` and `A` → `- A + B * / D E C`	final combine

✅ Prefix result: - * + A B C / D E

🧩 Summary for (A + B) * C - D / E: Infix: (A + B) * C - D / E Postfix: A B + C * D E / - Prefix: - * + A B C / D E

⚙️ What is it used for?

Infix — everyday math, programming languages (C, Python, Java)
Postfix — stack‑based evaluation, HP calculators, compilers (bytecode), expression trees
Prefix — Lisp / Scheme, functional languages, compilers (S‑expression), minimal parentheses

🔄 conversion: Infix → Postfix (Shunting‑yard) & Infix → Prefix (parse tree). Postfix & Prefix are operator‑precedence free — ideal for computer evaluation.

🧠 Why does it matter? Postfix & Prefix eliminate the need for parentheses and precedence rules — they are directly evaluable with a stack. Compilers often convert infix expressions to postfix or prefix during code generation. Also used in expression trees, reverse‑Polish calculators, and functional programming (prefix).

📌 quick comparison infix (A+B)×C → needs precedence postfix AB+C× → stack eval prefix ×+ABC → recursive eval