Hausdorff dimension of the harmonic measure on trees

For a large class of Markov operators on trees we prove the formula HD\,ν=h/l connecting the Hausdorff dimension of the harmonic measure ν on the tree boundary, the rate of escape l and the asymptotic entropy h. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations.