Why do we need prefix, postfix notation

Postfix notation, also known as RPN, is very easy to process left-to-right. An operand is pushed onto a stack; an operator pops its operand(s) from the stack and pushes the result. Little or no parsing is necessary. It's used by Forth and by some calculators (HP calculators are noted for using RPN).

Prefix notation is nearly as easy to process; it's used in Lisp.

Infix notation is easy to read for humans, whereas pre-/postfix notation is easier to parse for a machine. The big advantage in pre-/postfix notation is that there never arise any questions like operator precedence.

For example, consider the infix expression 1 # 2 $ 3. Now, we don't know what those operators mean, so there are two possible corresponding postfix expressions: 1 2 # 3 $ and 1 2 3 $ #. Without knowing the rules governing the use of these operators, the infix expression is essentially worthless.

Or, to put it in more general terms: it is possible to restore the original (parse) tree from a pre-/postfix expression without any additional knowledge, but the same isn't true for infix expressions.