  We study the totally asymmetric simple exclusion process (TASEP) on trees
where particles are generated at the root. Particles can only jump away from
the root, and they jump from $x$ to $y$ at rate $r_{x,y}$ provided $y$ is
empty. Starting from the all empty initial condition, we show that the
distribution of the configuration at time $t$ converges to an equilibrium. We
study the current and give conditions on the transition rates such that the
current is of linear order or such that there is zero current, i.e. the
particles block each other. A key step, which is of independent interest, is to
bound the first generation at which the particle trajectories of the first $n$
particles decouple.
