Extend the argument given in the proof of Lemma 10.4.1 to show that a tree with more than one vertex has at least two vertices of degree 1.

To prove:

A tree with more than one vertex has at least two vertices of degree 1.

Given:

The lemma, any tree that has more than one vertex has at least one vertex of degree 1.

Proof:

Let T be a tree with more than 1 vertex.

Let v0 be a vertex of T and let e0 be an edge incident on v. Such an edge e needs to exist, because T contains more than 1 vertex and a tree cannot contain any isolated vertices (as a tree needs to be connected).

While deg(v0)>1, choose an edge e′ that is incident on v such that e′≠e (which exists as deg(v0)>1 ) and let v′ be the vertex on the other end of the edge e′ (which exists as a tree contains no loops).

Let then v0=v′ and e0=e′.

Repeat until you obtain that deg(v0)=1 (which needs to occur as the tree is finite).

v0 is then, first vertex of degree 1