Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / R / 1 / c

Within each connected component with $n_i$ vertices, grow a tree by starting at one vertex and repeatedly adding an edge to an unreached vertex. Connectedness guarantees such an edge until every vertex has been reached. Exactly $n_i-1$ edges are added, so the component contains at least that many edges. Summing over the $k$ components givesThe selected trees form a spanning forest, and equality holds precisely when the original graph itself is a forest.