  We provide a new representation of an $\mathbb R$-tree by using a special set
of metric rays. We have captured the four-point condition from these metric
rays and shown an equivalence between the $\mathbb R$-trees with radial and
river metrics, and these sets of metric rays. In stochastic analysis, these
graphical representation theorems are of particular interest in identifying
Brownian motions indexed by $\mathbb R$-trees.
