sentences:  Tree-graded space
  
  A geodesic metric space formula_1 is called tree-graded space, with respect to a collection of connected proper subsets called "pieces", if any two distinct pieces intersect by at most one point, and every non-trivial simple geodesic triangle of formula_1 is contained in one of the pieces.Thus, for pieces of bounded diameter, tree-graded spaces behave like real trees in their coarse geometry (in the sense of Gromov), while allowing non-tree-like behavior within the pieces.