  Let $1 \le r < n$ be integers. We give a proof that the group
$\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, \sigma_{n}})$ of automorphisms of
the one-sided shift on $n$ letters embeds naturally as a subgroup
$\mathcal{H}_{n}$ of the outer automorphism group
$\mathop{\mathrm{Out}}({G_{n,r}})$ of the Higman-Thompson group $G_{n,r}$. From
this, we can represent the elements of
$\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, \sigma_{n}})$ by finite state
non-initial transducers admitting a very strong synchronizing condition.
  Let $H \in \mathcal{H}_{n}$ and write $|H|$ for the number of states of the
minimal transducer representing $H$. We show that $H$ can be written as a
product of at most $|H|$ torsion elements. This result strengthens a similar
result of Boyle, Franks and Kitchens, where the decomposition involves more
complex torsion elements and also does not support practical \textit{a priori}
estimates of the length of the resulting product.
  We also explore the number of foldings of de Bruijn graphs and give a
counting result for these for word length $2$ and alphabet size $n$.
  Finally, we offer new proofs of some known results about
$\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, \sigma_{n}})$.
