Let X be a curve that arises as p-cyclic cover of the p-adic line and denote by X_R its stable reduction. We start by giving an overview of what is known about the model X_R. Later we discuss new methods to compute the number of cycles in the special fiber X_k of X_R and certain components of positive genus. In particular we get a criterion for when X_k is tree-like and for X to be a Mumford curve. The talk will be illustrated by various examples.