Some equivalent characterizations of metric trees can be extracted from Theorem 8 of the paper "Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces" by Dress (1984) [DOI link](https://doi.org/10.1016/0001-8708(84)90029-X). The focus of the theorem is more on subsets of metric trees and injective hulls/tight spans, but the proof is self-contained.