Tree graded space

A geodesic metric space X 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 X 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 Druţu & Sapir (2005) in their study of the asymptotic cones of hyperbolic groups.