Introduction to Postfix Notation

Postfix notation, also known as Reverse Polish Notation (RPN), is a method of writing arithmetic expressions where the operator follows the operands. It eliminates the need for parentheses to define the order of operations, making it especially useful in computer science for stack-based calculations.

For example, the infix expression A + B becomes AB+ in postfix notation.

Expression Tree Basics

An expression tree is a binary tree structure used to represent expressions. Each internal node represents an operator, and each leaf node represents an operand. Traversing an expression tree collects the elements of the original expression in a specific notation—usually infix, prefix, or postfix.

The Role of Postfix Notation in Expression Trees

Postfix notation is used to simplify the construction of expression trees. Its inherent order makes it straightforward to build an expression tree using a stack-based approach:

1. Operands are pushed onto a stack.
2. When an operator is encountered, the required number of operands are popped from the stack.
3. A subtree is formed with the operator as the root, and the operands as the children.
4. The subtree is then pushed back onto the stack, treated as a single operand for further operations.

Step-by-step Conversion from Postfix Notation to an Expression Tree

Example

Consider the postfix expression AB+C*.

1. Initialize an empty stack.
2. Process each symbol:
   • Symbol: A
     • Operand, push onto stack. Stack: [A]
   • Symbol: B
     • Operand, push onto stack. Stack: [A, B]
   • Symbol: +
     • Operator, pop the top two operands (B and A).
     • Create a new tree node with the operator as the root: +
     • Subtree:
     • Push the subtree back onto the stack.
   • Symbol: C
     • Operand, push onto stack.
   • Symbol: *
     • Operator, pop the top two (subtree and operand C).
     • Create a new tree node with the operator as the root: *
     • Subtree:
     • Push the result back onto the stack.

• Parentheses Elimination: Postfix notation does not require parentheses, resolving ambiguity and simplifying parsing.
• Efficient Compilation: Many compilers use postfix for code generation as it is closer to machine-level instructions.
• Stack-based Evaluation: It aligns naturally with stack operations, which are fundamental in computer programming and execution.