Abstract: It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete -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. Moreover, this metric measures how far the trees are from being rooted isometric.
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