  We give a generalization to a continuous setting of the classic Markov chain
tree Theorem. In particular, we consider an irreducible diffusion process on a
metric graph. The unique invariant measure has an atomic component on the
vertices and an absolutely continuous part on the edges. We show that the
corresponding density at $x$ can be represented by a normalized superposition
of the weights associated to metric arborescences oriented toward the point
$x$. The weight of each oriented metric arborescence is obtained by the
exponential of integrals of the form $\int\frac{b}{\sigma^2}$ along the
oriented edges time a weight for each node determined by the local orientation
of the arborescence around the node time the inverse of the diffusion
coefficient at $x$. The metric arborescences are obtained cutting the original
metric graph along some edges.
