Tree-graded space

A geodesic metric space $$X$$ 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 $$X$$ is contained in one of the pieces.

If the pieces have 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 in their study of the asymptotic cones of hyperbolic groups.