  Asymptotic subcone of an unbounded metric space is another metric space,
capturing the structure of the original space at infinity. In this paper we
define a functional metric space S which is an asymptotic subcone of the
hyperbolic plane. This space is a real tree branching at every its point.
Moreover, it is a homogeneous metric space such that any real tree with
countably many vertices can be isometrically embedded into it. This implies
that every such tree is also an asymptotic subcone of the hyperbolic plane.
