Representability of Trees and Some of Their Applications

We prove that if a tree is representable as the free product of a finite set of cyclic groups of order two, then it is necessarily a Caley tree. For other trees, their presentations as some finite sets of sequences constructed from some recurrence relations are described. Using these presentations, we give a complete description of translation-invariant measures and a class of periodic Gibbs measures for a nonhomogeneous Ising model on an arbitrary tree. A sufficient condition for a random walk in a random environment on an arbitrary tree to be transient is described.