  We show that the metric structure of morphisms $f\colon Y\to X$ between
quasi-smooth compact Berkovich curves over an algebraically closed field admits
a finite combinatorial description. In particular, for a large enough skeleton
$\Gamma=(\Gamma_Y,\Gamma_X)$ of $f$, the sets $N_{f,\ge n}$ of points of $Y$ of
multiplicity at least $n$ in the fiber are radial around $\Gamma_Y$ with the
radius changing piecewise monomially along $\Gamma_Y$. In this case, for any
interval $l=[z,y]\subset Y$ connecting a rigid point $z$ to the skeleton, the
restriction $f|_l$ gives rise to a $profile$ piecewise monomial function
$\varphi_y\colon [0,1]\to[0,1]$ that depends only on the type 2 point
$y\in\Gamma_Y$. In particular, the metric structure of $f$ is determined by
$\Gamma$ and the family of the profile functions $\{\varphi_y\}$ with
$y\in\Gamma_Y^{(2)}$. We prove that this family is piecewise monomial in $y$
and naturally extends to the whole $Y^{\mathrm{hyp}}$. In addition, we extend
the theory of higher ramification groups to arbitrary real-valued fields and
show that $\varphi_y$ coincides with the Herbrand's function of
$\mathcal{H}(y)/\mathcal{H}(f(y))$. This gives a curious geometric
interpretation of the Herbrand's function, which applies also to non-normal and
even inseparable extensions.
