The critical orbit structure of quadratic polynomials in $\mathbb{Z}_p$

We study the forward orbit of the critical point for polynomials of the form $f_c=z^2+c$ defined over $\mathbb{Z}_p$. Hubbard trees capture the dynamical behavior for such maps with finite critical orbit in $\mathbb{C}$. We suggest a notion of Hubbard trees in the non-Archimedean setting, and describe the possible structures that arise for polynomials in $\mathbb{Z}_p$. As an example, we take a closer look at the dynamics of $f_c$ for $c\in\mathbb{Z}_3$.