  A factor graph of a point process is a graph whose vertices are the points of
the process, and which is constructed from the process in a deterministic
isometry-invariant way. We prove that the d-dimensional Poisson process has a
one-ended tree as a factor graph. This implies that the Poisson points can be
given an ordering isomorphic to the usual ordering of the integers in a
deterministic isometry-invariant way. For d \geq 4 our result answers a
question posed by Ferrari, Landim and Thorisson. We prove also that any
isometry-invariant ergodic point process of finite intensity in Euclidean or
hyperbolic space has a perfect matching as a factor graph provided all the
inter-point distances are distinct.
