Real tree

A real tree or R-tree is a metric space (M,d) such that for any x, y in M there is a unique arc from x to y, i.e., a continuous map f from an interval [a,b] to M such that f(a)=x and f(b)=y, and this arc is an isometric embedding.

There is a theory of group actions on R-trees.

Examples

• Each discrete tree can be regarded as an R-tree by a simple construction such that neighboring edges have distance one.
• The Paris metric makes the plane into an R-tree. If two points are on the same ray in the plane, their distance is defined as the Euclidean distance. Otherwise, their sum is define as the sum of the Euclidian distances of these two points to the origin.

References

• M. Bestvina (1999). R-trees in topology, geometry, and group theory.