Here you can change between infix (seen normally in most writing) and post fix also known as reverse polish notation online tool. To reduce the complexity of expression evaluation Prefix or Postfix To begin conversion of Infix to Postfix expression, first, we should know. Infix, Postfix and Prefix notations are three different but equivalent ways of writing You can use a similar trick to convert to and from parse trees – each.

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This type of notation is referred to as infix since the operator is in between the two operands that it is working on.

Which operands do they work on? The expression seems ambiguous. In fact, you have been reading and writing these types of expressions for a long time and they do not cause you any problem. Each operator has a precedence level. Operators of higher precedence are used before operators of lower precedence. The only thing that can change that order is the presence of parentheses.

The precedence order for arithmetic operators places multiplication and division above addition and subtraction. If two operators of equal precedence appear, then a left-to-right ordering or associativity is used.

B and C are multiplied first, and A is then added to that result. Although all this may be obvious to you, remember that computers need to know exactly what operators to perform postfxi in what order. One way to write an expression that guarantees there will be no confusion with respect ho the order of operations is to create what is called a fully parenthesized expression.

This type of expression uses one pair of parentheses for each operator. The parentheses dictate the order of operations; there is no ambiguity. There is also no need to remember any precedence rules.

There are two other very important expression formats that may not seem obvious to you at first. What would happen if we moved the operator before the two operands? Likewise, we could move the operator to the end.

These look a bit strange. These changes to the position of the operator with respect to the operands create two new expression formats, prefix and postfix.

Infix, Postfix and Prefix

Prefix expression notation requires that all operators precede the two operands that they work on. Postfix, on the other hand, requires that its operators come after the an operands. A few more examples should help to make this a bit clearer prefjx Table 2.

The prefjx operator then appears before the A and the result of the multiplication. Although the operators moved and now appear either before or after their respective operands, the order of the operands stayed exactly the same relative to one another. Recall that in this case, infix requires the parentheses to force the performance of the addition before the multiplication. The result of this operation becomes the first operand for the multiplication. The multiplication can be done to that result and the remaining operand C.

Consider these three expressions again see Table 3. Something very important has happened. Where did the parentheses go?

The answer is that the operators are no longer ambiguous with respect to the operands that they work on. Only infix notation requires the additional symbols. The order of operations within prefix and postfix pregix is completely determined by the position of the operator and nothing else. In many ways, this makes infix the least desirable notation to use. Table 4 shows some additional examples of infix expressions and the equivalent prefix and postfix expressions.


Be sure that you understand how they are equivalent in terms of the order of the operations being performed. So far, we have used ad hoc methods to convert between infix expressions and the equivalent prefix and postfix expression notations. As you might expect, there are algorithmic ways to perform the conversion that allow any expression of any complexity to be correctly transformed.

The first technique that we will consider postfx the notion of a fully parenthesized expression that was discussed earlier.

On closer observation, however, you can see that each parenthesis pair also denotes the beginning and the end of an operand pair with the corresponding operator in the middle.

If annd addition operator were also moved to its corresponding right parenthesis position and the matching left parenthesis were removed, the complete postfix expression would result see Figure 6. If we do the same thing but instead of moving the symbol to the position of the right parenthesis, we move it to the left, we get prefix notation see Figure 7. The position of the parenthesis pair is actually a clue to conversiin final position of the enclosed operator.

So in order to convert conversioh expression, no matter how complex, to either prefix or postfix notation, fully parenthesize the expression using prefjx order of operations.

Then move the enclosed operator to the position of either infixx left or the right parenthesis depending vonversion whether you want prefix or postfix notation. Here is a more complex expression: Figure 8 shows the conversion to postfix and prefix notations. We need to develop an algorithm to convert any infix expression to a postfix expression. To do this we will look closer at the conversion process.

We have already noted that the operands A, B, and C stay in their relative positions. It is only the operators that change position. The order of the operators in the original expression is reversed in the resulting postfix expression.

As we process the expression, the operators have to be saved somewhere since their corresponding right operands are not seen yet. Also, the order of these saved operators may need to be reversed due to their precedence. This is the case with the addition and the multiplication in this example. Since the addition operator comes before the multiplication operator and has lower precedence, it needs to appear after the multiplication operator is used.

Because of this reversal of order, it makes sense to consider using a stack to keep the operators until they are needed. We can now start to see how the conversion algorithm will work. When we see a left parenthesis, we will save it to denote that another operator of high precedence postfox be coming.

That operator will need to wait until the corresponding right parenthesis appears to denote its position recall the fully parenthesized technique. When that right parenthesis does appear, the operator can be popped from the stack. As we scan the infix expression from left to right, we will use a stack to keep the operators.

This will provide the reversal that we noted in the first example. The pfefix of the stack convrsion always be the most recently saved operator. Whenever we read a new operator, we will need to consider how that operator compares in precedence with the operators, if any, already on the stack.


Infix, Postfix and Prefix

Assume the infix expression is a string of tokens delimited by spaces. The operand tokens inifx the single-character identifiers A, B, C, and so on. The following steps will produce a string of tokens in postfix order.

In order to code the algorithm in Python, we will use a dictionary called prec to hold the precedence values for the operators. This dictionary will map each operator to an integer that can be compared against the precedence levels of other operators we have arbitrarily used the integers 3, 2, and 1.

The left parenthesis will receive the lowest value possible. This way any operator that is compared against it will have higher precedence and will be placed on top of it. Line 15 defines the operands to be any upper-case character or digit.

The complete conversion function is shown in ActiveCode 1. Converting Infix Expressions to Postfix Expressions intopost. As a final stack example, we will consider the evaluation of an expression that is already in postfix notation. In this case, a stack is again the data structure of choice. However, as you scan the postfix ifix, it is the operands that must wait, not the operators as in the conversion algorithm above. Conversoon way to think about the solution is that whenever an operator is seen on the input, the two most recent operands will be used in the evaluation.

As you scan the expression from left to right, you first encounter the operands 4 and 5. At this point, you are still unsure what to do with them until you see the next symbol. Placing each on the stack ensures that they are available if an cconversion comes next. In this case, the next symbol is another operand. So, as before, push it and check the next symbol. This means that the two most recent operands need to be used in a multiplication operation.

By popping the stack twice, we can get the proper operands and then perform the multiplication in this case getting the result We can now handle this result by placing it back on the stack so that it can be used as an operand for the later operators in the expression. When the final operator is processed, there will be only one value left on the stack. Pop and return it as the result of the expression. Figure 10 shows the stack contents as this entire example expression is being processed.

There are two things to note in this example. First, the stack size grows, shrinks, and then grows again as the subexpressions are evaluated. Second, the division operation needs to be handled carefully. Recall that the operands in the postfix expression are in their original order since postfix changes only the placement of operators.

Postfis the operands for the division are popped from the stack, they are reversed. Assume the postfix expression is a string of tokens delimited by spaces. The output will be an integer result. The complete function for the evaluation of postfix expressions is shown in ActiveCode 2.

Last modified: July 31, 2020