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Infix, Postfix and Prefix notations are three different but equivalent ways of writing expressions. It is easiest to demonstrate the differences by looking at examples of operators that take two operands.
 Infix notation: X + Y
 Operators are written inbetween their operands. This is the usual way we
write expressions. An expression such as
A * ( B + C ) / D
is usually taken to mean something like: "First add B and C together, then multiply the result by A, then divide by D to give the final answer."  Postfix notation (also known as "Reverse Polish notation"): X Y +
 Operators are written after their operands. The infix expression given
above is equivalent to
A B C + * D /
 The order of evaluation of operators is always lefttoright, and brackets cannot be used to change this order. Because the "+" is to the left of the "*" in the example above, the addition must be performed before the multiplication.
 Operators
act on values immediately to the left of them. For example, the "+" above uses
the "B" and "C". We can add (totally unnecessary) brackets to make this
explicit:
( (A (B C +) *) D /)
 Thus, the "*" uses the two values immediately preceding: "A", and the result of the addition. Similarly, the "/" uses the result of the multiplication and the "D".
 Prefix notation (also known as "Polish notation"): + X Y
 Operators are written before their operands. The expressions given above
are equivalent to
/ * A + B C D
 As for Postfix, operators are evaluated lefttoright and brackets are
superfluous. Operators act on the two nearest values on the right. I have
again added (totally unnecessary) brackets to make this clear:
(/ (* A (+ B C) ) D)
Infix notation needs extra information to make the order of evaluation of
the operators clear: rules built into the language about operator precedence
and associativity, and brackets ( )
to allow users to
override these rules. For example, the usual rules for associativity say that
we perform operations from left to right, so the multiplication by A is
assumed to come before the division by D. Similarly, the usual rules for
precedence say that we perform multiplication and division before we perform
addition and subtraction.
Although Prefix "operators are evaluated lefttoright", they use values to their right, and if these values themselves involve computations then this changes the order that the operators have to be evaluated in. In the example above, although the division is the first operator on the left, it acts on the result of the multiplication, and so the multiplication has to happen before the division (and similarly the addition has to happen before the multiplication).
Because Postfix operators use values to their left, any values involving computations will already have been calculated as we go lefttoright, and so the order of evaluation of the operators is not disrupted in the same way as in Prefix expressions.
In all three versions, the operands occur in the same order, and just the
operators have to be moved to keep the meaning correct. (This is particularly
important for asymmetric operators like subtraction and division:
A  B
does not mean the same as
B  A
; the former is equivalent to
A B 
or  A B
, the latter to
B A 
or  B A
).
The order of the operators in a postfix expression can be (but doesn't have to be) the opposite of their order in the corresponding prefix expression. 
Infix  Postfix  Prefix  Notes 

A * B + C / D 
A B * C D / + 
+ * A B / C D 
multiply A and B, divide C by D, add the results 
A * (B + C) / D 
A B C + * D / 
/ * A + B C D 
add B and C, multiply by A, divide by D 
A * (B + C / D) 
A B C D / + * 
* A + B / C D 
divide C by D, add B, multiply by A 
The most straightforward method is to start by inserting all the implicit brackets that show the order of evaluation e.g.:
Infix  Postfix  Prefix 

( (A * B) + (C / D) )
 ( (A B *) (C D /) +)
 (+ (* A B) (/ C D) )

((A * (B + C) ) / D)
 ( (A (B C +) *) D /)
 (/ (* A (+ B C) ) D)

(A * (B + (C / D) ) )
 (A (B (C D /) +) *)
 (* A (+ B (/ C D) ) ) 
You can convert directly between these bracketed forms simply by moving the operator within the brackets e.g.
(X + Y)
or
(X Y +)
or
(+ X Y)
.Repeat this for all the operators in an expression, and finally remove any superfluous brackets.
You can use a similar trick to convert to and from parse trees  each bracketed triplet of an operator and its two operands (or subexpressions) corresponds to a node of the tree. The corresponding parse trees are:
/ * + / \ / \ / \ * D A + / \ / \ / \ * / A + B / / \ / \ / \ / \ A B C D B C C D ((A*B)+(C/D)) ((A*(B+C))/D) (A*(B+(C/D)))
Because Infix is so common in mathematics, it is much easier to read, and so is used in most computer languages (e.g. a simple Infix calculator). However, Prefix is often used for operators that take a single operand (e.g. negation) and function calls.
Although Postfix and Prefix notations have similar complexity, Postfix is
slightly easier to evaluate in simple circumstances, such as in some calculators
(e.g. a simple
Postfix calculator), as the operators really are evaluated strictly
lefttoright (see note above).
For lots more information about notations for expressions, see my CS2111 lectures.
Page Created: Thursday, 13November2014... 5:42:19 AM Latest Update: Wednesday, 19July2017... 6:53 AM
Herb O. Buckland
herbobuckland@hotmail.com