A geodesic metric space X {\displaystyle 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 {\displaystyle X} is contained in one of the pieces. Tree-graded spaces behave like real trees "up to what can happen within the pieces", while allowing non-tree-like behavior within the pieces.
A geodesic metric space X {\displaystyle 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 {\displaystyle X} is contained in one of the pieces. Tree-graded spaces behave like real trees "up to what can happen within the pieces", while allowing non-tree-like behavior within the pieces.
Source: Wikipedia "Tree-graded space" · CC BY-SA 4.0
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