Let $ G $ be a graph other than the trivial one, with exactly one vertex of degree 1. Show that $ G $ cannot be a tree.
Proof. Since we know that every tree has $ n-1 $ edges, then the total degree of any tree must be $ 2 (n-1) $. Then there are $ (n-1) $ vertices with degree $ \geq 2 $ while only one vertex of degree 1. Thus, adding to find the total of the vertices, we have that the total degree of the vertices is $ \geq 2 (n-1) + 1 = 2n-1 $, which is a contradiction. Therefore $ G $ is not a tree. $ \square $