On the number of registers needed to evaluate arithmetic expressions

The question of how many temporary storage registers are needed to evaluate compiled arithmetic and masking expressions is discussed. It is assumed that any combination of left-to-right, right-to-left, top-to-bottom, and bottom-to-top techniques may be used to evaluate an expression, but that no factoring or re-arranging of the expression may occur. On this basis, the maximum number of registers needed to evaluate nonparenthesized expressions isN+1, withN the number of dyadic operator precedence levels. For parenthesized expressions with a maximum ofK nested parenthetical subexpressions, the maximum number of registers needed is (K+1)N+1.