  We study the enumeration problem for different kind of tree parking functions
introduced recently, called tree parking functions, tree parking distributions,
prime tree parking functions, and prime tree parking distributions, for rooted
labelled trees of important combinatorial tree families including labelled
ordered, unordered and binary trees. Using combinatorial decompositions of the
underlying structures yields, after solving the resulting equations, implicit
characterizations of suitable generating functions of the total number of such
tree parking functions for trees of size $n$ and $n$ successful drivers, from
which we obtain exact and asymptotic enumeration results. The approach can be
extended to the general situation of tree parking functions for trees of size
$n$ and $m<n$ drivers for which we are also able to characterize the generating
functions solutions, which allow, by applying analytic combinatorics
techniques, a study of the asymptotic behaviour of the total number of tree
parking functions and distributions for $n \to \infty$ depending on the load
factor $0 < \alpha = \frac{m}{n} < 1$.
