  Suppose that $m$ drivers each choose a preferred parking space in a linear
car park with $n$ spots. In order, each driver goes to their chosen spot and
parks there if possible, and otherwise takes the next available spot if it
exists. If all drivers park successfully, the sequence of choices is called a
parking function. Classical parking functions correspond to the case $m=n$; we
study here combinatorial and probabilistic aspects of this generalized case.
  We construct a family of bijections between parking functions $\text{PF}(m,
n)$ with $m$ cars and $n$ spots and spanning forests $\mathscr{F}(n+1, n+1-m)$
with $n+1$ vertices and $n+1-m$ distinct trees having specified roots. This
leads to a bijective correspondence between $\text{PF}(m, n)$ and monomial
terms in the associated Tutte polynomial of a disjoint union of $n-m+1$
complete graphs. We present an identity between the "inversion enumerator" of
spanning forests with fixed roots and the "displacement enumerator" of parking
functions. The displacement is then related to the number of graphs on $n+1$
labeled vertices with a fixed number of edges, where the graph has $n+1-m$
disjoint rooted components with specified roots.
  We investigate various probabilistic properties of a uniform parking
function, giving a formula for the law of a single coordinate. As a side result
we obtain a recurrence relation for the displacement enumerator. Adapting known
results on random linear probes, we further deduce the covariance between two
coordinates when $m=n$.
