  We study the maximum out forests of a (weighted) digraph and the matrix of
maximum out forests. A maximum out forest of a digraph G is a spanning subgraph
of G that consists of disjoint diverging trees and has the maximum possible
number of arcs. If a digraph contains any out arborescences, then maximum out
forests coincide with them. We provide a new proof to the Markov chain tree
theorem saying that the matrix of Ces`aro limiting probabilities of an
arbitrary stationary finite Markov chain coincides with the normalized matrix
of maximum out forests of the weighted digraph that corresponds to the Markov
chain. We discuss the applications of the matrix of maximum out forests and its
transposition, the matrix of limiting accessibilities of a digraph, to the
problems of preference aggregation, measuring the vertex proximity, and
uncovering the structure of a digraph.
