Real valued functions and metric spaces quasi-isometric to trees
  We prove that if X is a complete geodesic metric space with uniformly generated first homology group and $f: X\to R$ is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree.   Using this and adapting the definition of hyperbolic approximation we obtain an intrinsic sufficent condition for a metric space to be PQ-symmetric to an ultrametric space. 
\'Alvaro Mart\'inez-P\'erez
math.GT