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

Representation Theorems of $\mathbb{R}$-trees and Brownian Motions Indexed by $\mathbb R$-trees