  We prove that the free uniform spanning forest of any bounded degree proper
plane graph is connected almost surely, answering a question of Benjamini,
Lyons, Peres and Schramm.
  We provide a quantitative form of this result, calculating the critical
exponents governing the geometry of the uniform spanning forests of transient
proper plane graphs with bounded degrees and codegrees. We find that the same
exponents hold universally over this entire class of graphs provided that
measurements are made using the hyperbolic geometry of their circle packings
rather than their usual combinatorial geometry.
