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Infix, Postfix and Prefix
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."
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.
(see CS2121 lecture).

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)
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
).
Examples:
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

Converting between these notations
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)))
Computer Languages
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.