It is well known that a pseudo-Anosov homeomorphism of a closed hyperbolic surface admits two invariant transverse measured laminations. The singularity properties of those laminations record important dynamical characteristics of the homeomorphism. More generally, the same is true for arbitrary points of the of the Teichmuller space. Recently, the study of the properties of “random” pseudo-Anosovs as well as of “random” points in the Thurston boundary, and the properties of the corresponding measured laminations, became an important topic in geometric topology. We undertake a similar study for free group automorphisms and the boundary of the Culler-Vogtmann Outer space. In that contexts pseudo-Anosovs are replaced by the so-called “fully irreducible” automorphisms. Points of the boundary of the Outer space are R-trees, equipped with (usually highly mixing) isometric free group actions that in various senses generalize measured laminations on surfaces. We discuss the branching properties of "random" automorphisms and "random" (in the sense of harmonic measures) trees in the boundary of the Outer space. The talk is based on joint papers with Catherine Pfaff, Joseph Maher and Samuel Taylor.