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We prove that if X is a complete geodesic metric space with uniformlygenerated first homology group and $f: X\to R$ is metrically proper on theconnected components and bornologous, then X is quasi-isometric to a tree.  Using this and adapting the definition of hyperbolic approximation we obtainan intrinsic sufficent condition for a metric space to be PQ-symmetric to anultrametric space.

Real valued functions and metric spaces quasi-isometric to trees