Trees generate topological spaces (having branches for points) metrizable by ultrametrics. Such assignments give rise to a functor from trees to ultrametrizable spaces using special morphisms. The author shows that if one takes locally finite trees of vertex degree at least three and equivalence classes of quasi-isometries on such trees, then the functor is faithful and its image consists of perfect compact ultrametric spaces and bi-Hölder homeomorphisms as morphisms. The paper contains many other useful results on ultrametric spaces and special trees and their morphisms. It is a revision of the author’s thesis at Vanderbilt University in 2006. In some sense, the study generalizes that of B. Hughes [Adv. Math. 189, No. 1, 148–191 (2004; Zbl 1061.57021)].