  Let $\phi : C' \to C$ be a finite morphism between smooth, projective,
irreducible curves defined over a non-archimedean valued, algebraically closed
field $k$. This morphism induces a morphism between the analytifications of the
curves. We will construct a compatible pair of deformation retractions of
$C'^{an}$ and $C^{an}$ whose images $\Upsilon_{C'^{an}}$ and
$\Upsilon_{C^{an}}$ are closed subspaces of $C'^{an}$ and $C^{an}$ which are
homeomorphic to finite metric graphs. We refer to such closed subspaces as
skeleta. In addition, the subspaces $C'^{an}$ and $C^{an}$ are such that their
complements in the two analytifications decompose into the disjoint union of
Berkovich open balls and annuli. To these skeleta we can associate a genus. The
pair of compatible deformation retractions forces the morphism $\phi^{an}$ to
restrict to a map $\Upsilon_{C'^{an}} \to \Upsilon_{C^{an}}$. We will study how
the genus of $\Upsilon_{C'^{an}}$ can be calculated using the morphism
$\phi^{an}: \Upsilon_{C'^{an}} \to \Upsilon_{C^{an}}$.
