  In this paper we construct spanning trees in hyperbolic graphs that represent
their hyperbolic compactification in a good way: so that the tree has a bounded
number of distinct rays to each boundary point. The bound depends only on the
(Assouad) dimension of the boundary. As a corollary we sharpen a result of
Gromov which says that from every hyperbolic graph with bounded degrees one can
construct a tree outside the graph with a continuous surjection from the ends
of the tree onto the hyperbolic boundary such that the surjection is
finite-to-one. We will construct a tree with these properties inside the
hyperbolic graph, which in addition is also a spanning tree of that graph.
