  An optimal transport path may be viewed as a geodesic in the space of
probability measures under a suitable family of metrics. This geodesic may
exhibit a tree-shaped branching structure in many applications such as trees,
blood vessels, draining and irrigation systems. Here, we extend the study of
ramified optimal transportation between probability measures from Euclidean
spaces to a geodesic metric space. We investigate the existence as well as the
behavior of optimal transport paths under various properties of the metric such
as completeness, doubling, or curvature upper boundedness. We also introduce
the transport dimension of a probability measure on a complete geodesic metric
space, and show that the transport dimension of a probability measure is
bounded above by the Minkowski dimension and below by the Hausdorff dimension
of the measure. Moreover, we introduce a metric, called "the dimensional
distance", on the space of probability measures. This metric gives a geometric
meaning to the transport dimension: with respect to this metric, the transport
dimension of a probability measure equals to the distance from it to any finite
atomic probability measure.
