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!