In the field of computer science, Polish notation (also known as prefix notation) is a mathematical notation in which operators are placed before their operands. This notation eliminates the need for parentheses to indicate the order of operations. To evaluate expressions written in Polish notation, a specific data structure called a stack is commonly used.

## Stack Data Structure

A stack is an abstract data type that follows the Last-In-First-Out (LIFO) principle. It can be visualized as a vertical stack of objects, where the last object placed on top is the first one to be removed.

To evaluate an expression in Polish notation, we use a stack to store intermediate results and operands. The algorithm for evaluating Polish notation expressions using a stack can be summarized as follows:

- Create an empty stack.
- Read tokens from left to right.
- If the token is an operand, push it onto the stack.
- If the token is an operator, pop two operands from the stack and perform the operation on them.
- Push the result back onto the stack.
- Repeat steps 3-5 until all tokens have been processed.
- The final result will be at the top of the stack.

This algorithm takes advantage of both the LIFO nature of stacks and Polish notation’s operator placement to evaluate expressions efficiently. By always performing operations on two operands at a time, it ensures correct evaluation of complex expressions without needing parentheses or any explicit order of operations rules.

## Example

Let’s consider an example expression in Polish notation: “+ * 5 – 6 3 4”.

We start with an empty stack:

- Stack: []

As we read the tokens from left to right:

- The first token is “+”, which is an operator. We pop two operands from the stack (5 and the result of evaluating the next subexpression) and perform the addition:
- Stack: [5]
- The next token is “*”, another operator. We pop two operands (6 and 3) and perform multiplication:
- Stack: [5, 18]
- We continue reading tokens and evaluating subexpressions until we reach the end:
- Stack: [5, 18, -2]
- Stack: [5, 16]
- Stack: [21]

The final result is at the top of the stack, which in this case is 21.

### Conclusion

In conclusion, to evaluate expressions written in Polish notation efficiently, we use a stack data structure. The stack allows us to store intermediate results and operands while maintaining proper order of operations without parentheses or explicit rules.

The algorithm for evaluating Polish notation expressions using a stack follows a simple process of pushing operands onto the stack and performing operations on pairs of operands. By understanding this data structure and algorithm, you can effectively evaluate expressions written in Polish notation in your own programs.