Prove that A tree with n vertices has (n - 1) edges.   

Ans: From the definition of a tree a root comprise indegree zero and all other nodes comprise indegree one. There should be (n - 1) incoming arcs to the (n - 1) non-root nodes. If there is any another arc, this arc should be terminating at any of the nodes. If the node is root, after that its indegree will become one and that is in contradiction along with the fact that root all time has indegree zero. If the end point of this extra edge is any non-root node after that its indegree will be two, which is once again a contradiction. Therefore there cannot be more arcs. Hence, a tree of n vertices will have exactly (n - 1) edges.