  We prove that, for certain extensions of valued fields which admit a sensible
theory of ramification groups, there exist canonical towers that correspond to
the break-points of their Herbrand function. In particular, each of the
intermediate field extensions in the tower has a Herbrand function with only
one break-point and there is at most one extension with trivial Herbrand
function.
  We apply the result to the setting of finite morphisms of Berkovich curves
where we prove the existence of canonical local and global factorization of
such morphisms according to their metric properties.
  Finally, we use the canonical factorizations to prove harmonicity properties
finite morphisms satisfy at each type 2 point: formulas that can be regarded as
a refinement of the Riemann-Hurwitz formula for such morphisms.
