Prove that if G is a Tree Graph then there must be at least

Prove that if G is a Tree Graph then there must be at least two nodes with degree 1.

Solution

Let G be a tree graph with number of vertices N (N>1). Since G is a tree graph the number of edges in G is N-1.

So the degree sum of the tree is 2(N-1).

Now this degree sum is due to the N vertices, therefore, 2(N-1) is to be distributed among the N vertices.

But since a tree is connected, so no vertex has degree 0, moreover, if each vertex contributes at least 1 to the degree sum then it remains N-2 degrees, and we have N vertices. Thus there must be two vertices which have degree exactly equal to 1.