On the tree-likeness of hyperbolic spaces

In proper hyperbolic geodetic spaces we construct rooted $\mathbb{R}$-trees with the following properties. On the one hand, every ray starting at the root is quasi-geodetic; so these $\mathbb{R}$-trees represent the space itself well. At the same time, the trees boundary reflects the boundary of the space in that the number of disjoint rays to a boundary point is bounded in terms of the (Assouad) dimension of the hyperbolic boundary.