Infix-to-Prefix Conversion Methods
Three approaches to the same conversion: how they work, why they exist, and when each one helps.
Context
Infix puts an operator between its operands: A + B. Prefix puts it first: + A B. Conversion changes the notation, while preserving the expression's grouping; it does not calculate a value.
This lesson covers binary operators + - * / ^, where ^ means exponentiation. Each operator takes two operands.
Why
Different tasks need different intermediate representations. A stack keeps pending work in order, a tree preserves relationships, and recursion follows nested parsing rules.
These are three useful approaches, rather than an exhaustive list or three mutually exclusive algorithms. A stack-based or recursive parser can build an expression tree.
Example
Use the same example for every method:
Infix
A + B * CPrefix
+ A * B CMultiplication binds more tightly than addition, so B and C form one subexpression.
Concept
Every correct method must preserve three rules: parentheses, precedence and associativity.
(A + B) * Cbecomes* + A B C.A - B - Cgroups left:- - A B C.A ^ B ^ Cgroups right:^ A ^ B C.
Step-by-step
1. Stack-based conversion
Why it exists: Operators sometimes need to wait until their operands are complete. Stacks keep those pending operators and completed expression fragments in order.
- Read tokens from left to right. Push operands onto an expression stack.
- Keep operators on a separate stack. Before pushing an operator, reduce higher-priority operators, and equal-priority ones when the incoming operator is left-associative.
- For each reduction, pop the right fragment, then the left fragment. Push a new fragment containing operator, left fragment, then right fragment.
- Opening parentheses mark a boundary. At a closing parenthesis, reduce operations inside that boundary and discard the pair. Reduce remaining operators at the end.
Example: Push A, +, B, *, C onto their respective stacks. Reduce * into * B C, then reduce + into + A * B C.
Main benefit: The push/pop trace makes each conversion decision visible. This is the method used by our calculator.
Tradeoff: You must apply precedence, associativity and operand order correctly. For A ^ B ^ C, keep the equal-priority ^ pending so the right operation reduces first.
2. Expression tree and preorder traversal
Why it exists: A notation string shows the result, but a tree keeps the expression's structure. That structure helps with visualization, evaluation and later transformations.
- Parse the infix expression into a tree that respects parentheses and operator rules.
- Put an operator at each internal node and an operand at each leaf.
- Visit the root first, then its left subtree, then its right subtree. This preorder traversal produces prefix notation.
Example: For A + B * C, + is the root. Its left child is A; its right child is * with children B and C. Visit +, A, *, B, C.
Main benefit: You can see exactly which operands belong to each operator. Our Visual Breakdown shows this structure after the stack conversion.
Tradeoff: A tree still needs a parsing algorithm to build it. It also stores nodes and relationships beyond the final prefix string.
3. Recursive parsing / precedence climbing
Why it exists: Parsers need to recognize nested expressions and control how tightly operators bind. Recursive calls can represent those nested parsing decisions directly.
- Read an operand or recursively parse a parenthesized expression.
- For each operator, use its precedence to decide which part belongs to its right operand.
- For a left-associative operator, parse the right side at a higher minimum precedence. For a right-associative operator such as ^, allow the same precedence.
- Combine operator, left prefix fragment and right prefix fragment as each call returns.
Example: For A + B * C, the right side of + includes B * C. That call returns * B C; the outer call returns + A * B C.
Main benefit: Precedence and associativity fit into a compact parsing routine. It is useful when conversion is part of a larger expression parser.
Tradeoff: A recursive call trace can be less familiar than a stack table. Deep nesting also needs attention to recursion limits.
Practice
Try (A + B) / (C - D). Which operator becomes the prefix root?
Show the answer
Division is the root: / + A B - C D. Each parenthesized group supplies one operand to division.
Where it’s used
Choose stack-based conversion for a visible push/pop walkthrough, an expression tree when you need to inspect or reuse structure, and recursive parsing when you are building a parser around precedence rules. None is universally best for every task.
The calculator uses stacks for conversion and a tree for its Visual Breakdown. Its step table shows the actual stack operations that produce the result.
Further reading: Theodore Norvell on expression parsing · Emory's shunting-yard explanation