Inside any proper hyperbolic geodesic space X we construct a rooted topological R-tree T that reflects the geometry of X in the following sense. All rays in T are quasi-geodesic in X. Every geodesic ray in X lies eventually close to a ray of T. The embedding of T in X extends continuously to their boundaries in a finite-to-one way, the number of boundary points of T mapping to a given boundary point of X being bounded if the (Assouad) dimension of the boundary of X is finite.