You are given a connected tree T with n nodes and a designated root r$,$ where every vertex v has a weight W$[$v$].$ A set of nodes S is a k$-$independent set of T if $|$S$|=$ k and no two nodes in S have an edge between them in T$.$ The weight of such a set is given by adding up the weights of all the nodes in S$,$ i$.$e$.$

W$($S$)=$ X

v in S

W$[$v$].$