  The purpose of this article is twofold. On one hand, we reveal the
equivalence of shift of finite type between a one-sided shift $X$ and its
associated hom tree-shift $\mathcal{T}_{X}$, as well as the equivalence in the
sofic shift. On the other hand, we investigate the interrelationship among the
comparable mixing properties on tree-shifts as those on multidimensional shift
spaces. They include irreducibility, topologically mixing, block gluing, and
strong irreducibility, all of which are defined in the spirit of classical
multidimensional shift, complete prefix code (CPC), and uniform CPC. In
summary, the mixing properties defined in all three manners coincide for
$\mathcal{T}_{X}$. Furthermore, an equivalence between irreducibility on
$\mathcal{T}_{A}$ and irreducibility on $X_A$ are seen, and so is one between
topologically mixing on $\mathcal{T}_{A}$ and mixing property on $X_A$, where
$X_A$ is the one-sided shift space induced by the matrix $A$ and $T_A$ is the
associated tree-shift. These equivalences are consistent with the mixing
properties on $X$ or $X_A$ when viewed as a degenerate tree-shift.
