Asymptotics in the schema of simple varieties of trees

We study the functional equation y(x) = xA(y(x)) satisfied by the generating functions in the schema of simple varieties of trees. The radii of convergence r, R of y, A respectively satisfy y(r) <= R. In the subcritical case (y(r) < R), y_n behaves asymptotically as C·r^(-n)·n^(-3/2). In the critical case (y(r)=R), we approach the problem of determining the asymptotics of a_n when the information of y(x) is well known. To that end, we give sufficient conditions that ensure that A(z) can be extended to a delta domain around its dominant singularity R, which is needed to be able to apply the transfer theorem when determining the asymptotics. We do a similar analysis for the equation y(x) = x+ B(y(x)), which appears in the additive schema of simple varieties of trees. The general framework we propose includes the three examples of extension to a delta domain of the functions A(z) and B(z) encountered in the literature so far