Geodesic
Representability of Trees and Some of Their Applications
Matematičeskie zametki, Tome 72 (2002) no. 4, pp. 516-527.

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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. 
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U. A. Rozikov. Representability of Trees and Some of Their Applications. Matematičeskie zametki, Tome 72 (2002) no. 4, pp. 516-527. http://geodesic.mathdoc.fr/item/MZM_2002_72_4_a4/

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