  We study random two-dimensional spanning forests in the plane that can be
viewed both in the discrete case and in their appropriately taken scaling
limits as a uniformly chosen spanning tree with some Poissonian deletion of
edges or points. We show how to relate these scaling limits to a stationary
distribution of a natural coalescent-type Markov process on a state-space of
abstract graphs with real-valued edge-weights. This Markov process can be
interpreted as a renormalization flow.
  This provides a model for which one can rigorously implement the formalism
proposed by the third author in order to relate the law of the scaling limit of
a critical model to a stationary distribution of such a renormalization/Markov
process: When starting from any two-dimensional lattice with constant
edge-weights, the Markov process does indeed converge in law to this stationary
distribution that corresponds to a scaling limit of UST with Poissonian
deletions.
  The results of this paper heavily build on the convergence in distribution of
branches of the UST to SLE$_2$ (a result by Lawler, Schramm and Werner) as well
as on the convergence of the suitably renormalized length of the loop-erased
random walk to the "natural parametrization" of the SLE$_2$ (a recent result by
Lawler and Viklund).
