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Let $K$ be a field, and let $f\in K(z)$ be rational function. The preimagesof a point $x_0\in P^1(K)$ under iterates of $f$ have a natural tree structure.As a result, the Galois group of the resulting field extensions of $K$naturally embed into the automorphism group of this tree. In unpublished workfrom 2013, Pink described a certain proper subgroup $M_{\ell}$ that thisso-called arboreal Galois group $G_{\infty}$ must lie in if $f$ is quadraticand its two critical points collide at the $\ell$-th iteration. Afterpresenting a new description of $M_{\ell}$ and a new proof of Pink's theorem,we state and prove necessary and sufficient conditions for $G_{\infty}$ to bethe full group $M_{\ell}$.

Arboreal Galois groups for quadratic rational functions with colliding critical points