Good reduction for endomorphisms of the projective line in terms of the branch locus

Let K be a number field and v a non archimedean valuation on K. We say that an endomorphism $\Phi:\mathbb{P}_1 \to \mathbb{P}_1$ has good reduction at v if there exists a model $\Psi$ for $\Phi$ such that $deg \Psi_v$, the degree of the reduction of $\Psi$ modulo v, equals $deg \Psi$ and $\Psi_v$ is separable. We prove a criterion for good reduction that is the natural generalization of a result due to Zannier in [Z3]. Our result is in connection with other two notions of good reduction, the simple and the critically good reduction. The last part of our article is dedicated to prove a characterization of the maps whose iterates, in a certain sense, preserve the critically good reduction.