  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.
