Location theory on networks has been widely investigated by researchers from different fields for more than thirty years due to its significance and pratical value. Among various location problems, the p-center and the p-median problems are the most common. The p-facility k-centrum problem, introduced by Slater, is a generalization of the above two problems. The objective is to minimize the sum of the k largest service distances from clients to their nearest servers. When p is an arbitrary integer, the problem is NP-hard on general networks. Therefore, most researchers have devoted to the single-facility case, i.e. p=1, or the case that the networks under consideration are trees. This paper focuses on the single-facility k-centrum problem on a tree. For this problem, Tamir had an O(nlog² n) time algorithm. In this paper, an O(nlog n) time algorithm with is proposed.