  In this paper we investigate the geometric properties of quasi-trees, and
prove some equivalent criteria. We give a general construction of a tree that
approximates the ends of a geodesic space, and use this to prove that every
quasi-tree is $(1,C)$-quasi-isometric to a simplicial tree. As a consequence,
we show that Gromov's tree approximation lemma for hyperbolic spaces can be
improved in the case of quasi-trees to give a uniform approximation for any set
of points, independent of cardinality. From this we show that having uniform
tree approximation for finite subsets is equivalent to being able to uniformly
approximate the entire space by a tree. As another consequence, we note that
the boundary of a quasi-tree is isometric to the boundary of its approximating
tree under a certain choice of visual metric, and that this gives a natural
extension of the standard metric on the boundary of a tree.
