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 $d \ge 4$ 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.
"Trees and Matchings from Point Processes." Electron. Commun. Probab. 8 17 - 27, 2003. https://doi.org/10.1214/ECP.v8-1066