It is known that the essential spectrum of a Schrödinger operator H on \ell^{2}(\mathbb{N})  is equal to the union of the spectra of right limits of H. The natural generalization of this relation to \mathbb{Z}^n  is known to hold as well. In this talk we study the possibility of generalizing this characterization of \sigma_{ess}(H)  to trees. We give indications for the failure of the general statement in this case, while presenting a natural family of models where it still holds. This is a joint work with Jonathan Breuer.