Given a tree $T$ of $n$ vertices which has exactly $3$ leaves prove that complement graph of the tree is not an Euler graph (has Euler circuit).
A tree with exactly $3$ leaves must have exactly one vertex of degree $3$ so as not to violate Handshaking lemma. $T$ also has exactly $3$ vertices of degree $1$ and may have $0$ or more vertices of degree $2$. For example let $T$ be:
Then $T$'s complement is a triangle and the degrees of all vertices are even so it's Euler!
I guess my thinking is wrong because I'm proving the opposite of what I'm supposed to prove. What am I missing?