Let, $G(V,E)$ be a graph with a weight function $w(v)$ defined for all vertices. We need to find a tree $T(V',E')$ such that the resulting tree have maximum sum of weights of vertices, where $V' \in V$ and $E' \in E$. In short, we need to remove some vertices(along with all their edges) from $G(V,E)$ such that the resulting graph is a tree and have maximum weight among all such trees. $T(V'.E')$ should be an induced subgraph of $G(V,E)$.
First, sort the vertices in descending order. Then, pick the vertices in that order ensuring that adding a vertex won't create cycles. If it creates cycles, don't consider that vertex. To check if adding a particular vertex adds cycles, I thought of assigning each component a uniaue number. Hence, if adding a vertex results in joining that vertex to two vertices of the same component, we don't add that vertex. But, I couldn't find an efficient way on how should we keep track of which vertex belongs of which component. Can DSU help me with this approach?