Tree-graded space

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is called a tree-graded space with respect to a collection of connected proper subsets called pieces, if any two distinct pieces intersect in at most one point, and every non-trivial simple geodesic triangle of

Tree-graded spaces were introduced by Cornelia Druţu and Mark Sapir (2005) in their study of the asymptotic cones of relatively hyperbolic groups.

This point of view allows for a notion of relative hyperbolicity that makes sense for geodesic metric spaces and which is invariant under quasi-isometries.

For instance, a CAT(0) group has isolated flats, if and only if all its asymptotic cones are tree-graded metric spaces all of whose pieces are isometric to euclidean spaces.