  In this paper, we explore three combinatorial descriptions of semistable
types of hyperelliptic curves over local fields: dual graphs, their quotient
trees by the hyperelliptic involution, and configurations of the roots of the
defining equation (`cluster pictures'). We construct explicit combinatorial
one-to-one correspondences between the three, which furthermore respect
automorphisms and allow to keep track of the monodromy pairing and the Tamagawa
group of the Jacobian. We introduce a classification scheme and a naming
convention for semistable types of hyperelliptic curves and types with a
Frobenius action. This is the higher genus analogue of the distinction between
good, split and non-split multiplicative reduction for elliptic curves. Our
motivation is to understand $L$-factors, Galois representations, conductors,
Tamagawa numbers and other local invariants of hyperelliptic curves and their
Jacobians.
