Let $T=(V, E')$ be a spanning tree of a graph $G=(V, E)$. Rather than verifying for any subset of vertices $S\subseteq V$ that $|E'(S)|=|S|-1$, is there an efficient way to satisfy the spanning tree constraint (as the complexity for calculating all vertices subsets is exponential)?
To confirm that $T$ is a tree, you need to confirm that (a) it is connected and (b) $\vert E'\vert = \vert V \vert - 1$. The second part is trivial. To confirm it is connected, start at an arbitrarily chosen node $v\in V$ and label all nodes adjacent to $v$ in $T$. For each freshly labeled node, label all unlabeled nodes adjacent to it and proceed recursively until you run out of nodes to process. If all nodes ended up labeled, $T$ is connected, and you are done.