Finding a core of a tree with pos/neg weight

Let T = (V,E) be a tree. A core of T is a path P , such that the sum of the weighted distances from all vertices to this path is minimized. In this paper we consider the semi-obnoxious case where the vertices have positive or negative weights. We prove that if the sum of the weights of vertices is negative the core must be a single vertex and when the sum of the weights of vertices is zero there exists a core which is a vertex. Morgan and Slater [Journal of Algorithms 1 (1980) 247] presented a linear time algorithm for finding the core of a tree with only positive weights of vertices. We show that their algorithm also works for semi-obnoxious problems.