  We investigate relations between the pseudo-orbit-tracing property,
topological stability and openness for tree-shifts. We prove that a tree-shift
is of finite type if and only if it has the pseudo-orbit-tracing property which
implies that the tree-shift is topologically stable and all shift maps are
open. We also present an example of a tree-shift for which all shift maps are
open but which is not of finite type. It also turns out, that if a
topologically stable tree-shift does not have isolated points then it is of
finite type.
