Least-squares approximation by a tree distance

Let T be a tree with vertex set V (T ) = {1, . . . , n} and with a positive weight associated with each edge. The tree distance between i and j is the weight of the ij-path. Given a symmetric, positive real valued function on V (T )×V (T ), we consider the problem of approximating it by a tree distance corresponding to T, by the least-squares method. The problem is solved explicitly when T is a path or a double-star. For an arbitrary tree, a result is proved about the nature of the least-squares approximation. Some properties of the incidence matrix of all the paths in the tree are proved and used. We also note similar results for the corresponding matrix of a directed graph and obtain a formula for the MoorePenrose inverse of the all-paths matrix.