Q16Data Structures

The problem demands the manual algorithmic conversion of the complex, human-readable infix mathematical expression A + (B * C - (D / E ^ F) * G) * H into its machine-readable postfix equivalent (Reverse Polish Notation). This systemic conversion process inherently relies on strict adherence to established mathematical operator precedence (where Exponentiation ^ > Multiplication/Division * / > Addition/Subtraction + -) and respects the overriding associativity dictated by parentheses.

Step-by-Step Conversion Procedure

To execute this conversion flawlessly, we systematically evaluate the expression from the deepest nested parentheses outward, mathematically shifting operators to the immediate right of their corresponding operands.

Step 1 (Deepest Nested Parentheses): We focus strictly on the innermost expression (D / E ^ F). Applying operator precedence, Exponentiation (^) executes before Division (/). First, convert E ^ F $\to$ E F ^. Next, integrate the division: D / (E F ^) $\to$ D E F ^ /. The overall expression transforms into: A + (B * C - [D E F ^ /] * G) * H.

Step 2 (Secondary Parentheses): We now resolve the broader bracketed expression (B * C - [D E F ^ /] * G). Here, Multiplication operations must execute before Subtraction. Convert the first multiplication: B * C $\to$ B C *. Convert the second multiplication: [D E F ^ /] * G $\to$ D E F ^ / G *. Now apply the subtraction to connect them: [B C *] - [D E F ^ / G *] $\to$ B C * D E F ^ / G * -. The overall expression now simplifies to: A + [B C * D E F ^ / G * -] * H.

Step 3 (Final Global Resolution): We resolve the remaining operators on the global level: A + [Complex Term] * H. Multiplication must take precedence over Addition. Convert the multiplication: [Complex Term] * H $\to$ B C * D E F ^ / G * - H *. Finally, apply the global addition operator: A + [Result] $\to$ A B C * D E F ^ / G * - H * +. This represents the final, correctly formatted postfix expression, ready for stack-based evaluation.