Using stacks to convert expressions between infix, prefix, and postfix notations.
Mathematical expressions can be written in three different notations: infix, prefix (Polish notation), and postfix (Reverse Polish Notation). Infix is the standard notation we use, e.g., `a + b`, where the operator is between the operands. Prefix places the operator before the operands, e.g., `+ a b`. Postfix places the operator after the operands, e.g., `a b +`. Postfix and prefix notations are useful because they do not require parentheses or operator precedence rules for evaluation, making them ideal for computer evaluation. A stack is the perfect data structure for converting between these forms and for evaluating postfix/prefix expressions. To evaluate a postfix expression, we scan it from left to right. When we see an operand, we push it onto the stack. When we see an operator, we pop the top two operands from the stack, perform the operation, and push the result back onto the stack. The final answer is the single value remaining on the stack. The conversion from infix to postfix is more involved and uses a stack to hold operators. We iterate through the infix expression, pushing operands to the output and operators to the stack, managing precedence rules (e.g., `*` and `/` have higher precedence than `+` and `-`) as we go.