  It is well known that a countable group admits a left-invariant total order
if and only if it acts faithfully on R by orientation preserving
homeomorphisms. Such group actions are special cases of group actions on simply
connected 1-manifolds, or equivalently, actions on oriented order trees. We
characterize a class of left-invariant partial orders on groups which yield
such actions, and show conversely that groups acting on oriented order trees by
order preserving homeomorphism admit such partial orders as long as there is an
action with a point whose stabilizer is left-orderable.
