Hurwitz trees and deformations of Artin-Schreier covers

Let $R$ be a complete discrete valuation ring of equal characteristic $p>0$. Given a $\mathbb{Z}/p$-Galois cover of a formal disc over $R$, one can derive from it a semi-stable model for which the specializations of branch points are distinct and lie in the smooth locus of the special fiber. The description leads to a combinatorial object which resembles a classical Hurwitz tree in mixed characteristic, which we will give the same name. The existence of a Hurwitz tree is necessary for the existence of a $\mathbb{Z}/p$-cover whose branching data fit into that tree. We show that the conditions imposed by a Hurwitz tree's structure are also sufficient. Using this, we improve a known result about the connectedness of the moduli space of Artin-Schreier curves of fixed genus.