Path partition problems on trees have found various applications. In this paper, we present an \(O(n \log n)\) time algorithm for solving the following variant of path partition problem: given a rooted tree of n nodes \(1, \ldots , n\) , where vertex i is associated with a weight \(w_i\) and a cost \(s_i\) , partition the tree into several disjoint chains \(C_1,\ldots ,C_k\) , so that the weight of each chain is no more than a threshold \(w_0\) and the sum of the largest \(s_i\) in each chain is minimized. We also generalize the algorithm to the case where the cost of a chain is determined by the \(s_i\) of the vertex with the highest rank in the chain, which can be determined by an arbitrary total order defined on all nodes instead of the value of \(s_i\) .