Parking on trees with a (random) given degree sequence and the Frozen configuration model

Consider a rooted tree on the top of which we let cars arrive on its vertices. Each car tries to park on its arriving vertex but if it is already occupied, it drives towards the root of the tree and parks as soon as possible. In this article, we establish a natural coupling between the parking process on trees with prescribed degrees and an oriented configuration model. As a consequence, we recover the location of the phase transition for parking on critical Bienaym\'e--Galton--Watson trees already proven by Curien and H\'enard, and Contat.