  We consider the existence of the topological entropy of shift spaces on a
finitely generated semigroup whose Cayley graph is a tree. The considered
semigroups include free groups. On the other hand, the notion of stem entropy
is introduced. For shift spaces on a strict free semigroup, the stem entropy
coincides with the topological entropy. We reveal a sufficient condition for
the existence of the stem entropy of shift spaces on a semigroup. Furthermore,
we demonstrate that the topological entropy exists in many cases and is
identical to the stem entropy.
