Reducing systems for very small trees

Abstract: We study very small trees from the point of view of reducing systems of free factors, which are analogues of reducing systems of curves for a surface lamination; a non-trivial, proper free factor $F \leq \FN$ reduces $T$ if and only if $F$ acts on some subtree of $T$ with dense orbits. We characterize those trees, called arational, which do not admit a reduction by any free factor: $T$ is arational if and only if either $T$ is free and indecomposable or $T$ is dual to a surface with one boundary component equipped with an arational measured foliation. To complement this result, we establish some results giving control over the collection of all factors reducing a given tree. As an application, we deduce a form of the celebrated Bestvina-Handel classification theorem for elements of $Out(\FN)$. We also include an appendix containing examples of very small trees. The results of this paper are used in Bestvina and Reynolds (2012), where we describe the Gromov boundary of the complex of free factors.