About: Tree-graded space

A geodesic metric space 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 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. Tree-graded spaces were introduced by Cornelia Druţu and Mark Sapir in their study of the asymptotic cones of hyperbolic groups.