  We introduce and study a variant of the Wasserstein distance on the space of
probability measures, specially designed to deal with measures whose support
has a dendritic, or treelike structure with a particular direction of
orientation. Our motivation is the comparison of and interpolation between
plants' root systems. We characterize barycenters with respect to this metric,
and establish that the interpolations of root-like measures, using this new
metric, are also root like, in a certain sense; this property fails for
conventional Wasserstein barycenters. We also establish geodesic convexity with
respect to this metric for a variety of functionals, some of which we expect to
have biological importance.
