We define a totally asymmetric simple exclusion process on trees, where the particles enter the tree from a reservoir attached to the root. Particles can only jump following edges on the tree and only if they are jumping away from the root, to a non-occupied vertex.

We are interested in two different aspects of this particle system. First we want to investigate the current of particles across a given generation at a specific time, when we start from an initially empty initial condition. Second, we would like some current estimates along a ray. Both answers depend heavily on our choice of jump rates for the particles and different regimes of behaviour will be presented during the talk.