Abstract
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 reccurence 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.
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Rozikov, U.A. Representability of Trees and Some of Their Applications. Mathematical Notes 72, 479–488 (2002). https://doi.org/10.1023/A:1020580227830
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DOI: https://doi.org/10.1023/A:1020580227830