Tokens stream in; the operator stack defers operators by precedence while the output queue builds postfix (RPN). Then the RPN is evaluated on a value stack.
^ (right-assoc) > * / > + - (left-assoc).
The output queue is exactly the postfix traversal of the parse tree; evaluating it left-to-right on a value stack yields the answer.
See junior.md for the algorithm and professional.md for the correctness proof.