  One of the main properties of modulus on graphs is Fulkerson duality. In this
paper, we study Fulkerson duality for spanning tree modulus. We introduce a new
notion of Beurling partition, and we identify two important ones, which
correspond to the notion of strength and maximum denseness of an arbitrary
graph. These special partitions, also give rise to two deflation processes that
reveal a hierarchical structure for general graphs. While Fulkerson duality for
spanning tree families can be deduced from a well-known result in combinatorics
due to Chopra, we give an alternative approach based on a result of
Nash-Williams and Tutte. Finally, we introduce the weighted variant of spanning
tree modulus.
