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. The focus of the theorem is more on subsets of metric trees and injective hulls/tight spans, but the proof is self-contained.

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. The focus of the theorem is more on subsets of metric trees and injective hulls/tight spans, but the proof is self-contained.