  We initiate the study of the forward and backward shifts on the Lipschitz
space of a tree, $\mathcal L$, and on the little Lipshitz space of a tree,
${\mathcal L}_0$. We determine that the forward shift is bounded both on
$\mathcal L$ and on ${\mathcal L}_0$ and, when the tree is leafless, it is an
isometry; we also calculate its spectrum. For the backward shift, we determine
when it is bounded on $\mathcal L$ and on ${\mathcal L}_0$, we find the norm
when the tree is homogeneous, we calculate the spectrum for the case when the
tree is homogeneous, and we determine, for a general tree, when it is
hypercyclic.
