  We study three Markov processes on infinite, unrooted, regular trees: the
stochastic Ising model (also known as the Glauber heat bath dynamics of the
Ising model), a majority voter dynamic, and a coalescing particle model. In
each of the three cases the tree exhibits a preferred direction encoded into
the model. For all three models, our main result is the existence of a
stationary but non-reversible measure. For the Ising model, this requires
imposing that the inverse temperature is large and choosing suitable
non-uniform couplings, and our theorem implies the existence of a stationary
measure which looks nothing like a low-temperature Gibbs measure. The
interesting aspect of our results lies in the fact that the analogous processes
do not have non-Gibbsian stationary measures on $\mathbb Z^d$, owing to the
amenability of that graph. In fact, no example of a stochastic Ising model with
a non-reversible stationary state was known to date.
