  Let $K$ be a field, and let $f\in K(z)$ be a rational function of degree
$d\geq 2$. The Galois group of the field extension generated by the preimages
of $x_0\in K$ under all iterates of $f$ naturally embeds in the automorphism
group of an infinite $d$-ary rooted tree. In some cases the Galois group can be
the full automorphism group of the tree, but in other cases it is known to have
infinite index. In this paper, we consider a previously unstudied such case:
that $f$ is a polynomial of degree $d=3$, and the two finite critical points of
$f$ collide at the $\ell$-th iteration, for some $\ell\geq 2$. We describe an
explicit subgroup $Q_{\ell,\infty}$ of automorphisms of the $3$-ary tree in
which the resulting Galois group must always embed, and we present sufficient
conditions for this embedding to be an isomorphism.
