  For every $p\in(0,\infty)$, a new metric invariant called umbel $p$-convexity
is introduced. The asymptotic notion of umbel convexity captures the geometry
of countably branching trees, much in the same way as Markov convexity, the
local invariant which inspired it, captures the geometry of bounded degree
trees. Umbel convexity is used to provide a "Poincar\'e-type" metric
characterization of the class of Banach spaces that admit an equivalent norm
with Rolewicz's property $(\beta)$. We explain how a relaxation of umbel
$p$-convexity, called umbel cotype $p$, plays a role in obtaining compression
rate bounds for coarse embeddings of countably branching trees. Local analogs
of these invariants, fork $q$-convexity and fork cotype $q$, are introduced and
their relationship to Markov $q$-convexity and relaxations of the $q$-tripod
inequality is discussed. The metric invariants are estimated for a large class
of Heisenberg groups. Finally, a new characterization of non-negative curvature
is given.
