  We are interested in measures of central tendency for a population on a
network, which is modeled by a metric tree. The location parameters that we
study are generalized Fr\'echet means obtained by minimizing the objective
function $\alpha \mapsto E[\ell(d(\alpha,X))]$ where $\ell$ is a generic convex
nondecreasing loss.
  We leverage the geometry of the tree and the geodesic convexity of the
objective to develop a notion of directional derivative in the tree, which
helps up locate and characterize the minimizers.
  Estimation is performed using a sample analog. We extend to a metric tree the
notion of stickiness defined by Hotz et al. (2013), we show that this
phenomenon has a non-asymptotic component and we obtain a sticky law of large
numbers. For the particular case of the Fr\'echet median, we develop
non-asymptotic concentration bounds and sticky central limit theorems.
