Illustration
To get a real feel for evaluating postfix expressions using a stack, let's walk through a simple example. Imagine we have the postfix expression 6 5 2 3 +  *. Here's how we'd evaluate it step by step:
Start with an empty stack: Imagine a clean slate, ready to be written on.

Scan 6: It's an operand, so push 6 onto the stack. Our stack now has one element: [6].

Scan 5: Another operand. Push 5 onto the stack, making it [6, 5].

Scan 2: Again, an operand. Our stack grows to [6, 5, 2].

Scan 3: Operand. The stack becomes [6, 5, 2, 3].

Scan +: An operator. Pop 3 & 2 (the top two elements) off the stack & add them. 2 + 3 = 5. Push this result back onto the stack, leaving us with [6, 5, 5].

Scan : Operator time. Pop the top two elements (5 & 5), subtract the second from the first (5  5 = 0), & push the result (0) back. The stack now looks like [6, 0].

Scan *: Last operator. Pop 0 & 6, multiply them (6 * 0 = 0), & push the result back. Our stack, now just [0], holds the final result.
So, the result of the postfix expression 6 5 2 3 +  * is 0. By breaking it down step by step, we can see how each part of the expression affects the stack & leads to the final answer.
Postfix Evaluation for Multidigit Numbers
When dealing with multidigit numbers in postfix expressions, the approach remains largely the same, but with a slight twist. The key difference is in how we interpret the numbers during the scanning phase. Here's how it goes:
Reading the Expression
As we scan the expression, we need to be mindful of spaces or delimiters that separate numbers. Unlike singledigit numbers, multidigit numbers can't be read one character at a time.
Pushing Numbers onto the Stack
Whenever we come across a number (now it could be more than one digit long), we read the entire number, treating spaces as the separator. This whole number is then pushed onto the stack as a single entity.
Operators Work the Same
When we encounter an operator, the process is the same as with singledigit expressions. We pop the top two numbers from the stack, perform the operation, and push the result back onto the stack.
For example, let's evaluate the postfix expression 23 8 7 + *. Here's the stepbystep breakdown:

Start with an empty stack.

Scan and push 23: The stack now has [23].

Scan and push 8: The stack becomes [23, 8].

Scan and push 7: Now, the stack is [23, 8, 7].

Encounter +: Pop 8 and 7, add them (8 + 7 = 15), and push 15 back. The stack updates to [23, 15].
Encounter *

Pop 23 and 15, multiply (23 * 15 = 345), and push the result 345 back onto the stack.

With the final stack being [345], we conclude that 345 is the result of the expression 23 8 7 + *.
This method extends the utility of postfix expressions to accommodate more complex and realistic numerical computations, making them even more powerful in practical applications.
Frequently Asked Questions
Why use postfix notation instead of the more common infix notation?
Postfix notation simplifies the computation process for computers. It eliminates the need for parentheses, making the order of operations clear & straightforward, which is not always the case with infix notation.
Can postfix evaluation handle operations beyond basic arithmetic like addition and subtraction?
Absolutely. Postfix evaluation is versatile & can handle a wide range of operations, including multiplication, division, exponentiation, & even more complex functions, as long as the operation involves taking a certain number of operands from the stack & pushing the result back.
How does the stack help in evaluating postfix expressions?
The stack provides an organized way to manage operands & operators. By following the "last in, first out" principle, it ensures that the most recent operands are used with the most recent operator, maintaining the correct order of operations without needing parentheses.
Conclusion
In this article, we've learned the essentials of evaluating postfix expressions, a fundamental concept that's instrumental in understanding computer arithmetic operations. Starting with the basics of using a stack for evaluation, we looked into the process step by step, even tackling multidigit numbers to show the method's applicability to more complex expressions.
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