Data structures are a fundamental concept in computer science and are essential for organizing and manipulating data efficiently. Infix and postfix notations are two methods of writing mathematical expressions, each with its own advantages and use cases.

In this article, we will explore what infix and postfix notations are and how they are used in data structures.

## What is Infix Notation?

Infix notation is the most commonly used method of writing mathematical expressions. It is the familiar way we write expressions, with operators placed between operands.

For example, in the expression 3 + 4, the operator (+) is placed between the operands (3) and (4). Similarly, in the expression 5 * (6 – 2), we have multiple operators (+, *, -) placed between operands (5, 6, 2).

Infix notation follows certain rules to determine the order of operations. These rules include parentheses to specify precedence and associativity of operators.

For example, in the expression 5 * (6 – 2), the subtraction operation inside parentheses is performed first because parentheses have higher precedence than multiplication.

## What is Postfix Notation?

Postfix notation, also known as Reverse Polish Notation (RPN), is an alternative method for writing mathematical expressions. Unlike infix notation where operators are placed between operands, postfix notation places operators after their corresponding operands.

For example, the infix expression 3 + 4 can be written as the postfix expression 3 4 +. Similarly, the infix expression 5 * (6 – 2) can be written as the postfix expression 5 6 2 – *.

Postfix notation eliminates the need for parentheses to specify precedence because it relies on a stack-based evaluation algorithm. The algorithm reads the expression from left to right and uses a stack to store operands and intermediate results.

When an operator is encountered, it pops the required number of operands from the stack, performs the operation, and pushes the result back onto the stack.

## Comparison of Infix and Postfix Notations

Both infix and postfix notations have their own advantages and use cases. Infix notation is intuitive for humans to read and write, but it requires parentheses to specify precedence in complex expressions.

On the other hand, postfix notation eliminates the need for parentheses and allows for straightforward evaluation using a stack-based algorithm.

Postfix notation is particularly useful in computer programs where evaluating mathematical expressions is a common task. It simplifies expression parsing and evaluation by eliminating the need for complex precedence rules.

Additionally, postfix notation can be easily implemented using stacks or other data structures.

### Example: Converting Infix to Postfix

Let’s consider an example to demonstrate how infix expressions can be converted into postfix expressions. Suppose we have the following infix expression:

(5 + 6) * (7 – 2)

To convert this expression into postfix notation, we follow these steps:

- Initialize an empty stack.
- Read each symbol from left to right.
- If the symbol is an operand (number), output it.
- If the symbol is an operator:
- If the stack is empty or contains an opening parenthesis on top, push the operator onto the stack.
- If the symbol has higher precedence than the operator on top of the stack, push it onto the stack.
- If the symbol has lower precedence than the operator on top of the stack, pop operators from the stack and output them until a lower precedence operator is encountered or the stack becomes empty. Then push the symbol onto the stack.
- If the symbol is a closing parenthesis, pop operators from the stack and output them until an opening parenthesis is encountered. Discard both parentheses.
- After all symbols are read, pop any remaining operators from the stack and output them.

Using these steps, we can convert our infix expression (5 + 6) * (7 – 2) into the postfix expression:

5 6 + 7 2 – *

### Conclusion

Infix and postfix notations are two methods of writing mathematical expressions. Infix notation is widely used and familiar to humans, while postfix notation eliminates the need for parentheses and enables efficient evaluation using stacks.

Understanding these notations is essential for working with data structures that involve mathematical expressions.

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