  We study traveling wave solutions to bistable differential equations on
infinite $k$-ary trees. These graphs generalize the notion of classical square
infinite lattices and our results complement those for bistable lattice
equations on $\mathbb{Z}$. Using comparison principles and explicit lower and
upper solutions, we show that wave-solutions are pinned for small diffusion
parameters. Upon increasing the diffusion, the wave starts to travel with
non-zero speed, in a direction that depends on the detuning parameter. However,
once the diffusion is sufficiently strong, the wave propagates in a single
direction up the tree irrespective of the detuning parameter. In particular,
our results imply that changes to the diffusion parameter can lead to a
reversal of the propagation direction.
