How to Convert Postfix Notation to Infix Notation Using Stack
Working from left to right, scan each character of the postfix expression, and take one of the following two actions.
If the character is an operand, push it to the stack.
If the character is an operator, pop the top value from the stack for its right operand and pop the next top value from the stack for its left operand. Finally, insert the operator between its operands, place the infix string within a set of parenthesis, and then push the infix string back to the stack.
Repeat the above scanning and subsequent actions until all postfix characters have been handled. If the postfix expression was valid, you should be left with a single element in the stack, which is the infix equivalent of the postfix expression.
Posfix to Infix Conversion Examples
The following are two examples showing how to apply the conversion process detailed in the previous section.
Example #1: 5 6 2 ^ 2 - *
5 6 2 ^ 2 - *
The first character scanned is "5", which is an operand, so push it to the stack.
| 5 | ||
| Stack | Infix String |
5 6 2 ^ 2 - *
The next character scanned is "6", which is an operand, so push it to the stack.
| 6 5 | ||
| Stack | Infix String |
5 6 2 ^ 2 - *
The next character scanned is "2", which is an operand, so push it to the stack.
| 2 6 5 | ||
| Stack | Infix String |
5 6 2 ^ 2 - *
The next character scanned is "^", which is an operator, so pop its two operands from the stack. Pop 2 from the stack for the right operand and then pop 6 from the stack for the left operand. Next, insert the "^" between its two operands to form the infix string 6 ^ 2, and then place that string inside a set of parenthesis.
| 5 | (6 ^ 2) | |
| Stack | Infix String |
Next, push the infix string (6 ^ 2) to the stack.
| (6 ^ 2) 5 | ||
| Stack | Infix String |
5 6 2 ^ 2 - *
The next character scanned is "2", which is an operand, so push it to the stack.
| 2 (6 ^ 2) 5 | ||
| Stack | Infix String |
5 6 2 ^ 2 - *
The next character scanned is "-", which is an operator, so pop its two operands from the stack. Pop 2 from the stack for the right operand and then pop (6 ^ 2) from the stack for the left operand. Next, insert the "-" between its two operands to form the infix string (6 ^ 2) - 2, and then place that string inside a set of parenthesis.
| 5 | ((6 ^ 2) - 2) | |
| Stack | Infix String |
Next, push the infix string ((6 ^ 2) - 2) to the stack.
| ((6 ^ 2) - 2) 5 | ||
| Stack | Infix String |
5 6 2 ^ 2 - *
The next character scanned is "*", which is an operator, so pop its two operands from the stack. Pop ((6 ^ 2) - 2) from the stack for the right operand and then pop 5 from the stack for the left operand. Next, insert the "*" between its two operands to form the infix string 5 * ((6 ^ 2) - 2), and then place that string inside a set of parenthesis.
(5 * ((6 ^ 2) - 2)) | ||
| Stack | Infix String |
Next, push the infix string (5 * ((6 ^ 2) - 2)) to the stack.
| (5 * ((6 ^ 2) - 2)) | ||
| Stack | Infix String |
Since we are done scanning characters, the remaining element in the stack ((5 * ((6 ^ 2) - 2))) becomes the result of the postfix to infix conversion.
Postfix notation: 5 6 2 ^ 2 - *Infix equivalent: (5 * ((6 ^ 2) - 2))
Example #2: 7 2 ^ 25 10 5 / + * 13 -
7 2 ^ 25 10 5 / + * 13 -
The first character scanned is "7", which is an operand, so push it to the stack.
| 7 | ||
| Stack | Infix String |
7 2 ^ 25 10 5 / + * 13 -
The next character scanned is "2", which is an operand, so push it to the stack.
| 2 7 | ||
| Stack | Infix String |
7 2 ^ 25 10 5 / + * 13 -
The next character scanned is "^", which is an operator, so pop its two operands from the stack. Pop 2 from the stack for the right operand and then pop 7 from the stack for the left operand. Next, insert the "^" between its two operands to form the infix string 7 ^ 2, and then place that string inside a set of parenthesis.
(7 ^ 2) | ||
| Stack | Infix String |
Next, push the infix string (7 ^ 2) to the stack.
| (7 ^ 2) | ||
| Stack | Infix String |
7 2 ^ 25 10 5 / + * 13 -
The next character scanned is "25", which is an operand, so push it to the stack.
| 25 (7 ^ 2) | ||
| Stack | Infix String |
7 2 ^ 25 10 5 / + * 13 -
The next character scanned is "10", which is an operand, so push it to the stack.
| 10 25 (7 ^ 2) | ||
| Stack | Infix String |
7 2 ^ 25 10 5 / + * 13 -
The next character scanned is "5", which is an operand, so push it to the stack.
| 5 10 25 (7 ^ 2) | ||
| Stack | Infix String |
7 2 ^ 25 10 5 / + * 13 -
The next character scanned is "/", which is an operator, so pop its two operands from the stack. Pop 5 from the stack for the right operand and then pop 10 from the stack for the left operand. Next, insert the "/" between its two operands to form the infix string 10 / 5, and then place that string inside a set of parenthesis.
| 25 (7 ^ 2) | (10 / 5) | |
| Stack | Infix String |
Next, push the infix string (10 / 5) to the stack.
| (10 / 5) 25 (7 ^ 2) | ||
| Stack | Infix String |
7 2 ^ 25 10 5 / + * 13 -
The next character scanned is "+", which is an operator, so pop its two operands from the stack. Pop (10 / 5) from the stack for the right operand and then pop 25 from the stack for the left operand. Next, insert the "+" between its two operands to form the infix string 25 + (10 / 5), and then place that string inside a set of parenthesis.
| (7 ^ 2) | (25 + (10 / 5)) | |
| Stack | Infix String |
Next, push the infix string (25 + (10 / 5)) to the stack.
| (25 + (10 / 5)) (7 ^ 2) | ||
| Stack | Infix String |
7 2 ^ 25 10 5 / + * 13 -
The next character scanned is "*", which is an operator, so pop its two operands from the stack. Pop (25 + (10 / 5)) from the stack for the right operand and then pop (7 ^ 2) from the stack for the left operand. Next, insert the "*" between its two operands to form the infix string (7 ^ 2) * (25 + (10 / 5)), and then place that string inside a set of parenthesis.
((7 ^ 2) * (25 + (10 / 5))) | ||
| Stack | Infix String |
Next, push the infix string ((7 ^ 2) * (25 + (10 / 5))) to the stack.
| ((7 ^ 2) * (25 + (10 / 5))) | ||
| Stack | Infix String |
7 2 ^ 25 10 5 / + * 13 -
The next character scanned is "13", which is an operand, so push it to the stack.
| 13 ((7 ^ 2) * (25 + (10 / 5))) | ||
| Stack | Infix String |
7 2 ^ 25 10 5 / + * 13 -
The next character scanned is "-", which is an operator, so pop its two operands from the stack. Pop 13 from the stack for the right operand and then pop ((7 ^ 2) * (25 + (10 / 5))) from the stack for the left operand. Next, insert the "-" between its two operands to form the infix string ((7 ^ 2) * (25 + (10 / 5))) - 13, and then place that string inside a set of parenthesis.
(((7 ^ 2) * (25 + (10 / 5))) - 13) | ||
| Stack | Infix String |
Next, push the infix string (((7 ^ 2) * (25 + (10 / 5))) - 13) to the stack.
| (((7 ^ 2) * (25 + (10 / 5))) - 13) | ||
| Stack | Infix String |
Since we are done scanning characters, the remaining element in the stack ((((7 ^ 2) * (25 + (10 / 5))) - 13)) becomes the result of the postfix to infix conversion.
Postfix notation: 7 2 ^ 25 10 5 / + * 13 -Infix equivalent: (((7 ^ 2) * (25 + (10 / 5))) - 13)