1. 5.2 Trees  A tree is a connected graph without any cycles.

2.  Theorem 5.2.1 A graph with n vertices is a tree if and only if it has n −1 edges and no cycles. Proof 1.Suppose first that the graph is a tree 2.removing an edge from a tree results in a graph having two components, each of which is without a cycle. 3.after we have removed n−1 edges, what remains is a graph with n components; that is, one without any more edges.

3. 1.suppose we have a set of n−1 edges 2.Starting with a graph consisting of n components that is, consisting of the n vertices and no edges 3.Add these n-1 edges one at a time 4.Since each added edge must be between vertices in different components (for otherwise it would result in a cycle), it follows that each added edge decreases the number of components by one. 5.Thus, after the (n−1)th edge has been added, the graph has only one component and no cycles; in other words, it is a tree.

4.  Proposition 5.2.1 Every tree has at least two leaves. 1.Let D be the sum of the degrees of all the vertices of a graph. And D = 2(n−1) when the graph is a tree 2.Suppose tree has only one leave ◎ leave(degree=1) ◎ nonleaf degree is at least 2(degree sum=2*(n-1) ) 3.The sum of the degrees is 1+2*(n-1) ≠2(n-1) 4.A contradiction Proof Sum of the degrees=2x(5-1)=8

5.  Lemma 5.2.1 Proof Corollary 2.7.2 implies that which is equivalent to the identity stated in the lemma.

6.  Proposition 5.2.2 (Cayley’s Theorem) There are trees on a vertex set of size n.

7. Proof N(B) denote the numberof elements of B t(n) denote the number of trees on a set of n vertices Let L i denote the set of trees on V for which vertex i is a leaf, i = 1,..., n. 1.is true when n = 1 and n = 2 2.assume that it is true whenever the vertex set is of size smaller than n 3.Now consider the vertex set V = {1,..., n}, where n > 2 Since every tree has at least one leaf, we obtain from the inclusion–exclusion rule that

8. where the final equality follows from Lemma 5.2.1.