  Using local detailed balance we rewrite the Kirchhoff formula for stationary
distribution of Markov jump processes in terms of a physically interpretable
tree-ensemble. We use that arborification of path-space integration to derive a
McLennan-tree characterization close to equilibrium, as well as to obtain
response formula for the stationary distribution in the asymptotic regime of
large driving. Graphical expressions of currents and of traffic are obtained,
allowing the study of various asymptotic regimes. Finally, we present how the
matrix-forest theorem gives a representation of quasi-potentials, as used e.g.
for computing excess work and heat in nonequilibrium thermal physics. A variety
of examples illustrate and explain the graph elements and constructions.
