  Let f(z) be a rational function of degree at least 2 with coefficients in a
number field K, and assume that the second iterate f^2(z) of f(z) is not a
polynomial. The second author previously proved that for any b in K, the
forward orbit O_f(b) contains only finitely many quasi-S-integral points. In
this note we give an explicit upper bound for the number of such points.
