  In this paper, we first concentrate on the possible values and dense property
of entropies for isotropic and anisotropic axial products of subshifts of
finite type (SFTs) on $\mathbb{N}^d$ and $d$-tree $\mathcal{T}_d$. We prove
that the entropies of isotropic and anisotropic axial products of SFTs on
$\mathbb{N}^d$ are dense in $[0,\infty)$, and the same result also holds for
anisotropic axial products of SFTs on $\mathcal{T}_d$. However, the result is
no longer true for isotropic axial products of SFTs on $\mathcal{T}_d$. Next,
motivated by the work of Johnson, Kass and Madden [16], and Schraudner [28], we
establish the entropy formula and structures for full axial extension shifts on
$\mathbb{N}^d$ and $\mathcal{T}_d$. Combining the aforementioned results with
the findings on the surface entropy for multiplicative integer systems [8] on
$\mathbb{N}^d$ enables us to estimate the surface entropy for the full axial
extension shifts on $\mathcal{T}_d$. Finally, we extend the results of full
axial extension shifts on $\mathcal{T}_d$ to general trees.
