Let $P(z)$ the ordinary generating function of plane rooted trees and let $F(z) = e^{P(z)}$. Use saddle-point estimates of large powers to determine asymptotic estimates for $[z^n]P(z)$ and $[z^n]F(z)$.
Hint: Use Lagrange inversion first to state $[z^n]P(z)$ and $[z^n]F(z)$ in a form that allows to use saddle-point estimates of large powers.
Remark: You can find the section about "saddle-point estimates of large powers" from the Flajolet & Sedgewick book (p.586,587,588) below.
I do not understand how to do this.
First off, here is the theorem of Lagrange Inversion that I know from the Flajolet & Sedgewick book (p. 732):
I also know that for $P(z)$ holds the equation
$$P(z) = z \frac{1}{1-P(z)},$$
so we can use the Lagrange Inversion on $P(z)$ by setting $y(z) = P(z)$ and $\phi(u) = \frac{1}{1-u}$. But I do not understand what this has to do with the saddle-point estimates of large powers. Could you please give me a hint?
Since $P(z)=z/(1-P(z))$, by $(12)$ and $(14)$ in the Lagrange Inversion Theorem, we have $$ [z^n ]P(z) = \frac{1}{n}[u^{n - 1} ]\frac{1}{{(1 - u)^n }} $$ and $$ [z^n ]F(z) = [z^n ]{\rm e}^{P(z)} = \frac{1}{n}[u^{n - 1} ]\frac{{{\rm e}^u }}{{(1 - u)^n }}, $$ respectively. We apply the "saddle-point estimate of large powers" with $A(z) = B(z) = 1/(1 - z)$, $n-1$ in place of $n$, and $\lambda=1$. We find $\zeta=1/2$ and $\xi=8$. Thus, $$ [z^n ]P(z) = \frac{1}{n}[u^{n - 1} ]\frac{1}{{1 - u}}\frac{1}{{(1 - u)^{n - 1} }} \sim \frac{1}{n}2\frac{{2^{n - 1} }}{{(1/2)^n \sqrt {2\pi (n - 1) \cdot 8} }} \sim \frac{{4^{n - 1} }}{{\sqrt \pi n^{3/2} }} $$ as $n\to +\infty$. Similarly, we apply the theorem with $A(z) =\mathrm{e}^z/(1 - z)$, $B(z) = 1/(1 - z)$, $n-1$ in place of $n$, and $\lambda=1$. We find $\zeta=1/2$ and $\xi=8$. Therefore, $$ [z^n ]F(z) = \frac{1}{n}[u^{n - 1} ]\frac{{{\rm e}^u }}{{1 - u}}\frac{1}{{(1 - u)^{n - 1} }} \sim \frac{1}{n}2{\rm e}^{1/2} \frac{{2^{n - 1} }}{{(1/2)^n \sqrt {2\pi (n - 1) \cdot 8} }} \sim \sqrt {\frac{{\rm e}}{\pi }} \frac{{4^{n - 1} }}{{n^{3/2} }} $$ as $n\to +\infty$. I leave you as an exercise to check that the conditions of the "saddle-point estimate of large powers" are met.