It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete R-trees. We define a map on pairs of PQ-symmetric ultrametric spaces which characterizes the branching of the space. We also show that when the ultrametric spaces are the corresponding end spaces, this map defines a metric between rooted geodesically complete simplicial trees with minimal vertex degree 3 in the same quasi-isometry class.

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... eover, this metric measures how far the trees are from being rooted isometric. 325 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 326Á. MARTÍNEZ-PÉREZ In [10] we defined bounded distortion equivalences and we proved that bounded distortion equivalences characterize PQ-symmetric homeomorphisms between certain classes of bounded, complete, uniformly perfect, ultrametric spaces which we called pseudo-doubling. This class includes those ultrametric spaces arising up to similarity as the end spaces of bushy simplicial trees. Bounded distortion equivalences can be seen from a geometrical point of view as a coarse version of quasiconformal homeomorphisms (see for example [1], [7] and [8] for a geometric approach to quasiconformal maps) where instead of looking at the distortion of the spheres with the radius tending to 0 we consider the distortion of all of them. Of central importance in the theory of quasiconformal mappings are Teichmüller spaces. The Teichmüller space is the set of Riemannian surfaces of a given quasiconformal type, and the Teichmüller metric measures how far are the spaces from being conformally equivalent. There is also an extensive literature on this; see for example [1] and [8] . The question we deal with in this paper is to see if something similar to a Teichmüller metric can be defined with these bounded distortion equivalences playing the role of the quasiconformal homeomorphisms. Here we consider the set of ultrametric spaces of a given PQ-symmetry type. What is obtained is a function (see section 3) which is not a metric in the general framework of pseudo-doubling ultrametric spaces because it fails to hold the triangle inequality. Nevertheless, adding a condition on the metrics, it is enough to characterize what we call here the branching of the space, which is a natural concept when looking at the ultrametric space as the boundary of a tree. Theorem 1.1. Let (U, d), (U , d ) be ultrametric spaces. If the metrics d, d are pseudo-discrete, then (U, U ) = 0 if and only if (U, d) and (U , d ) have the same branching. As we mentioned before, if the ultrametric spaces are end spaces of R-trees, then the PQ-symmetry type of the boundary corresponds to the quasi-isometry type of the trees. In the case of a quasi-isometry class, [(R, z)], of rooted geodesically complete simplicial trees with minimal vertex degree 3, we prove Theorem 1.2. The function is an unbounded metric on [(R, z)]. This means that in the category of rooted simplicial trees with minimal vertex degree 3, the branching is enough to characterize the isometry type, the quasiisometry type and the "distance" between quasi-isometric objects. Preliminaries on trees, end spaces, and ultrametrics In this section, we recall the definitions on trees and their end spaces that are relevant to this paper. We also describe a well-known correspondence between trees and ultrametric spaces. See Feȋnberg [5] for an early result along these lines and Hughes [9] for additional background. Definition 2.