Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a Certain Property

Question

First, define a sequence $F_0,F_1,\dots$ of functions by $$F_0(x,z) = z,$$ $$F_k(x,z)=x\exp\left(F_{k-1}(x,z)\right) \quad \text{for }k\geq1.$$ So, for example, $$F_1(x,z) = x e^z, \quad F_2(x,z)=xe^{xe^z},\dots$$ etc. Also, set $F_{-1}(x,z)=0$. Now, let $$G(x,z) = \sum_{k=0}^\infty \left(F_k(x,z) - F_{k-1}(x,x)\right).$$ That is, $$G(x,z) = z + \left(xe^z - x\right)+\left(xe^{xe^z} - xe^x\right) + \dots$$

What I would like to do is to get some information (it doesn't have to be amazingly strong information...) about the asymptotics of the coefficient of the $x^{n-j}z^j$ term in the power series for $G(x,z)$.

Question: Does anyone know whether I have any hope in extracting any information from this generating function? If so, any ideas about what I should do/try? Even a pointer to something in the literature which might help me would be great!

By the way, the function $G(x,z)$ is closely linked to the Lambert $W$ Function. In particular, (I think) it is not hard to see that $$G(x,x)=\sum_{n=1}^\infty \frac{n^{n-1} x^n}{n!}$$ and it is well known that this function is the solution to the functional equation $$G(x,x) = x\exp(G(x,x)).$$ The thing that makes this question tricky therefore seems to be the presence of the second variable, $z$.

Remark: By the way, the coefficient of $x^{n-j}z^j$ in $$F_k(x,z)-F_{k-1}(x,x)$$ counts the number $n$-vertex trees rooted at vertex $1$ of height exactly $k$ such that there are exactly $j$ vertices at distance $k$ from the root. Therefore, the coefficient of $x^{n-j}z^j$ in $G(x,z)$ is the number of $n$-vertex trees (of any height) in which there are $j$ vertices at maximum distance from vertex $1$. If anyone knows anything about the number of such trees (independently of the generating function), then that would also be useful!

Answer 1

The asymptotic distribution of the number of nodes at maximal height in a random tree is known.

The following was known as "Wilf's conjecture" (H. S. Wilf stated it in 1991, evidently unaware of the fact that it had been popularized among Russian combinatorialists by V. F. Kolchin earlier):

Choose a labeled tree of $n$ vertices uniformly among all $n^{n-2}$ labeled trees and let $L_n$ be the number of vertices at maximal graph distance from its root (the vertex with label $1$).
Then $L_n$ converges in distribution as $n\rightarrow \infty$, i.e., there exists a probability distribution $(q_\ell)_{\ell\geq 1}$ such that $$\mathbb{P}(L_n=\ell)\longrightarrow q_\ell,\;\;\ell\ge 1$$

This conjecture was settled by Kesten and Pittel in 1996, see http://www.dx.doi.org/10.1002/(SICI)1098-2418(199607)8:4<243::AID-RSA1>3.0.CO;2-Y

Kesten and Pittel proved (Corollary 2, page 8) that the conjecture is true, and that $$q_\ell=\pi_\ell e^{-\ell}$$ where $(\pi_\ell)_{\ell\geq 1}$ is the unique solution of $$\pi_\ell\geq 0,\;\;\;\;\pi_\ell=\sum_{k=1}^\infty \pi_k\,e^{-k}\frac{k^\ell}{\ell!},\;\;\;\,\sum_{\ell=1}^\infty \pi_{\ell}e^{-\ell} =1$$