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{\bf Richard W. Kenyon and David B. Wilson}
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{\bf Combinatorics of Tripartite Boundary Connections for Trees and Dimers}
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A grove is a spanning forest of a planar graph in which every
component tree contains at least one of a special subset of vertices
on the outer face called nodes.  For the natural probability measure
on groves, we compute various connection probabilities for the nodes
in a random grove.  In particular, for ``tripartite'' pairings of the
nodes, the probability can be computed as a Pfaffian in the entries of
the Dirichlet-to-Neumann matrix (discrete Hilbert transform) of the
graph.  These formulas generalize the determinant formulas given by
Curtis, Ingerman, and Morrow, and by Fomin, for parallel pairings.
These Pfaffian formulas are used to give exact expressions for
reconstruction: reconstructing the conductances of a planar graph from
boundary measurements.  We prove similar theorems for the double-dimer
model on bipartite planar graphs.

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