Periodic oscillations of coefficients of power series that satisfy functional equations, a practical revision

For the solutions $\mathrm{\Phi }(z)$ of functional equations $\mathrm{\Phi }(z)=P(z)+\mathrm{\Phi }(Q(z))$, we derive a complete asymptotic of power series coefficients. As an application, we improve significantly an asymptotic of the number of $2,3$-trees with $n$ leaves given in Adv. Math. 44:180-205, 1982 by Andrew M. Odlyzko. The methods we consider can be applied to more general functional equations too.