Trees. Let G be a graph. Then G is connected if there is a path between every pair of vertices in G. We call G a tree if it is connected and does not contain a cycle. Prove the following theorem by using mathematical induction. Theorem 1. Let T be a tree on n vertices. Then T has n - 1 edges. In your proof, you may want to make use of the following lemma which you can assume to be correct and do not have to prove. Lemma 2. Let G be a connected graph, and let e be an edge in G that is not contained in any cycle. Then, the graph obtained from G by deleting e contains precisely two components