  In this paper we develop the metric theory for the outer space of a free
product of groups. This generalizes the theory of the outer space of a free
group, and includes its relative versions. The outer space of a free product is
made of $G$-trees with possibly non-trivial vertex stabilisers. The strategies
are the same as in the classical case, with some technicalities arising from
the presence of infinite-valence vertices.
  In particular, we describe the Lipschitz metric and show how to compute it;
we prove the existence of optimal maps; we describe geodesics represented by
folding paths. We show that train tracks representative of irreducible (hence
hyperbolic) automorphisms exist and that their are metrically characterized as
minimal displaced points, showing in particular that the set of train tracks is
closed. We include a proof of the existence of simplicial train tracks map
without using Perron-Frobenius theory.
  A direct corollary of this general viewpoint is an easy proof that relative
train track maps exist in both the free group and free product case.
