 Solution

Take root one, its children two and three, the children of two to be four and five, and those of three to be six and seven. Its edge set is $\{12,13,24,25,36,37\}$. The root has degree two, vertices two and three have degree three, and the remaining vertices have degree one.

Removing the smallest available leaves in order gives $4,5,2,1,6$, with recorded neighbors $2,2,1,3,3$. Therefore its Prüfer code is