  Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of
a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is
perfect and that $\mathrm{char}\ k > 2g+1$. Let $S = \mathrm{Spec}\ R$. Let $X$
be the minimal proper regular model of $C$ over $S$. Let $\mathrm{Art}\ (C/K)$
denote the Artin conductor of the $S$-scheme $X$ and let $\nu (\Delta_C)$
denote the minimal discriminant of $C$. We prove that $-\mathrm{Art}\ (C/K)
\leq \nu (\Delta_C)$. The key ingredients are a combinatorial refinement of the
discriminant introduced in this paper (called the metric tree) and a recent
refinement of Abhyankar's inversion formula for studying plane curve
singularities. We also prove combinatorial restrictions for $-\mathrm{Art}\
(C/K) = \nu (\Delta_C)$.
