= Solution
{c}
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
$$
\boxed{(2,2,1,3,3).}
$$
\Image[/past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2008-rooted-binary-tree.png]
{title=A labelled seven-vertex rooted binary tree with root one}
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