This paper considers a generalization of the inverse 1-median problem, the inverse k-centrum problem, on trees with variable vertex weights. In contrast to the linear time solvability of the inverse 1-median problem on trees, we prove that the inverse k-centrum problem on trees is NP-hard. Particularly, the inverse 1-center problem, a special case of the mentioned problem with k = 1, on a tree with n vertices can be solved in O(n^2) time.