  It is known that the essential spectrum of a Schr\"odinger operator $H$ on
$\ell^{2}\left(\mathbb{N}\right)$ 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 paper we generalize the notion of right
limits to general infinite connected graphs and construct examples of graphs
for which the essential spectrum of the Laplacian is strictly bigger than the
union of the spectra of its right limits. As these right limits are trees, this
result is complemented by the fact that the equality still holds for general
bounded operators on regular trees. We prove this and characterize the
essential spectrum in the spherically symmetric case.
