Enumeration of fully parked trees

We enumerate a class of fully parked trees. In a probabilistic context, this means computing the partition function $F(x,y)$ of the parking process where an i.i.d. number of cars arrives at each vertex of a Galton-Watson tree with a geometric offspring distribution, conditioned to leave no vertex unoccupied at the end. The variables $x$ and $y$ count the number of vertices in the tree and the number of cars exiting from the root, respectively. For any car arrival distribution $\mathbf b$, we obtain an explicit parametric expression of $F(x,y)$ in terms of the probability generating function $B(y)$ of $\mathbf b$. We show that the model has a generic phase where the singular behavior of $F(x,y)$ is essentially independent of $B(y)$, and a non-generic phase where it depends sensitively on the singular behavior of $B(y)$. The non-generic phase is further divided into two cases, which we call dilute and dense. We give a simple algebraic description of the phase diagram, and, under mild additional assumptions on $\mathbf b$, carry out detailed singularity analysis of $F(x,y)$ in the generic and the dilute phases. The singularity analysis uses the classical transfer theorem, as well as its generalization for bivariate asymptotics. In the process, we develop a variational method for locating the dominant singularity of the inverse of an analytic function, which is of independent interest. The phases defined in this paper are closely related to, but not the same as, the phases in the transition of macroscopic runoff described in arxiv:1912.06012 and related works.