  We study the topological entropy of hom tree-shifts and show that, although
the topological entropy is not a conjugacy invariant for tree-shifts in
general, it remains invariant for hom tree higher block shifts. In
doi:10.1016/j.tcs.2018.05.034 and doi:10.3934/dcds.2020186, Petersen and Salama
demonstrated the existence of topological entropy for tree-shifts and
$h(\mathcal{T}_X) \geq h(X)$, where $\mathcal{T}_X$ is the hom tree-shift
derived from $X$. We characterize a necessary and sufficient condition when the
equality holds for the case where $X$ is a shift of finite type. In addition,
two novel phenomena have been revealed for tree-shifts. There is a gap in the
set of topological entropy of hom tree-shifts of finite type, which makes such
a set not dense. Last but not least, the topological entropy of a reducible hom
tree-shift of finite type is equal to or larger than that of its maximal
irreducible component.
