Let G be a connected graph with n vertices and n −1 edges. We have to show that G contains no cycles. Assume to the contrary that G contains cycles.Remove an edge from a cycle so that the resulting graph is again connected. Continue this process of removing one edge from one cycle at a time till the resulting graph H is a tree. As H has n vertices, so number of edges in H is n−1. Now, the number of edges in G is greater than the number of edges in H. So n−1 > n−1, which is not possible. Hence, G has no cycles and therefore G is a tree.
