 Solution

Take a root $r$ with children $a,b$. Give $a$ the children $c,1$, give $c$ the children $2,3$, and give $b$ the children $4,5$. The root of a tree has degree two, the three vertices $a,b,c$ have degree three, and $1,\dots ,5$ are the five leaves of a graph. There are nine vertices and eight edges, as required by the handshaking lemma and the edge count for a tree.