On the average number of registers required for evaluating arithmetic expressions

Let An be the average number of registers required for evaluating arithmetic expressions of size n, or, equivalently, the minimal stack needed for exploring binary trees with n nodes. We give explicit expressions for An and related quantities and show that: An = log4(n) + C + E(log4n) + o(1) where C = 1/2 - γ + 2/2 log2 + log2Π 0.292 and E is continuous, periodic with period 1, with average value 0 and amplitude less than .05.