Bilateral parking procedures

We introduce the class of bilateral parking procedures on the integers. These generalize the classical one as follows: Instead of having cars systematically park in the nearest available spot on their right, we also allow cars to use the nearest spot to their left. We will distinguish the subclass of local procedures, which has the striking property that for any procedure of the class, the number of corresponding parking functions of length r is given by $(r + 1)^{r−1}$. We show how to extend definitions and results to probabilistic procedures, in which the decision to park left or right is random. We finally describe how bilateral procedures can naturally be encoded by labeled binary forests, whose combinatorics explain and unify several results from the literature