It is well known that each tree metric M has a unique realization as a tree, and that this realization minimizes the total length of the edges among all other realizations of M. We extend this result to the class of symmetric matrices M with zero diagonal, positive entries, and such that m_ij + m_kl <= max { m_ik + m_jl, m_il + m_jk } for all distinct i, j, k, l.