In this talk we deal with some lifting-to-characteristic-zero problems. Let R be a complete DVR in mixed characteristic, and let k be its residue field. We study a finite group G of automorphisms of R[[T]] together with the reduction of its action over k[[t]]. To deal with wild ramification phenomena in this context, the notion of Hurwitz tree has been introduced and worked out in the last ten years. This combinatorial object encodes at the same time the geometry of fixed points and the ramification theory of the G-action. We show in this talk how Hurwitz trees can be characterized in the setting of Berkovich spaces. We will explain how this result sheds new light on the local lifting problem and in which sense these embedded Hurwitz trees "parametrize" certain deformations of torsors.