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.
A tree with more than one vertex has at least two vertices of degree 1.
The lemma, any tree that has more than one vertex has at least one vertex of degree 1.
Let T be a tree with more than 1 vertex.
Let be a vertex of T and let 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 , choose an edge that is incident on v such that (which exists as ) and let be the vertex on the other end of the edge (which exists as a tree contains no loops).
Let then and .
Repeat until you obtain that (which needs to occur as the tree is finite).
is then, first vertex of degree 1
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