Parking on a random rooted binary tree

In this paper, we investigate the parking process on a uniform random rooted binary tree with $n$ vertices. Viewing each vertex as a single parking space, a random number of cars independently arrive at and attempt to park on each vertex one at a time. If a car attempts to park on an occupied vertex, it traverses the unique path on the tree towards the root, parking at the first empty vertex it encounters. If this is not possible, the car exits the tree at the root. We shall investigate the limit of the probability of the event that all cars can park when $\lfloor \alpha n \rfloor$ cars arrive, with $\alpha > 0$. We find that there is a phase transition at $\alpha_c = 2 - \sqrt{2}$, with this event having positive limiting probability when $\alpha < \alpha_c$, and the probability tending to 0 as $n \rightarrow \infty$ for $\alpha > \alpha_c$. This is analogous to the work done by Goldschmidt and Przykucki (arXiv:1610.08786) and Goldschmidt and Chen (arXiv:1911.03816), while agreeing with the general result proven by Curien and H\'enard (arXiv:2205.15932).