  Let $(A_u : u \in \mathbb{B})$ be i.i.d.~non-negative integers that we
interpret as car arrivals on the vertices of the full binary tree $
\mathbb{B}$. Each car tries to park on its arrival node, but if it is already
occupied, it drives towards the root and parks on the first available spot. It
is known that the parking process on $ \mathbb{B}$ exhibits a phase transition
in the sense that either a finite number of cars do not manage to park in
expectation (subcritical regime) or all vertices of the tree contain a car and
infinitely many cars do not manage to park (supercritical regime). We
characterize those regimes in terms of the law of $A$ in an explicit way. We
also study in detail the critical regime as well as the phase transition which
turns out to be "discontinuous".
