The author deals with the comparison of the following two fundamental metrical properties of a network: (a) it is a tree if and only if its metric is convex, and (b) it decomposes into finitely many treelike segments on which the metric is convex. He shows that (a) and (b) are intimately related proving that equivalence (a) holds in a class of networklike so-called reticular metric spaces that are characterized exactly by their decomposability into treelike segments.