Asymptotic expansions for the coefficients of analytic generating functions

Pairs F(x), G(x) of analytic generating functions that satisfy relations such as 1+G(x)=exp(F(x)) are studied. It is shown that, if F(x) satisfies fairly mild regularity conditions, such as those imposed by Hayman in his study of coefficients of some general classes of functions, then G(x) satisfies the much stricter conditions imposed by Harris and Schoenfeld. This enables one to obtain complete asymptotic expansions for the coefficients of G(x). Applications of this result are made to enumerations of trees.