Hyperbolicity is invariant under quasi-isometry

For geodesic metric spaces, a quasi-isometry preserves the property of being a Gromov-hyperbolic metric space. Images of metric geodesic sides are uniform quasigeodesics. The Morse lemma for quasi-geodesics keeps them uniformly close to metric geodesic sides in the hyperbolic target. Thinness there and the coarse lower distance bound pull a uniform triangle-thinness constant back to the source. Coarse surjectivity supplies a quasi-inverse for the reverse implication. Consequently a metric geodesic space quasi-isometric to a tree is hyperbolic, with no local finiteness assumption on the tree.