Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / R / 1 / iii

If there are $i$ nonroot interior vertices and $\ell$ leaves, then $1+i+\ell =7$ and the tree degree sum is $2+3i+\ell =12$. Hence $i=2$ and $\ell =4$. Distribute these two nonroot interior vertices among the root's two descendant branches.

Either each branch contains one interior vertex, producing the balanced tree with two cherries, or both are in one branch while the other branch is a leaf. The latter gives a single unbalanced shape, because left and right children are not ordered. Thus there are two unlabelled rooted binary trees. A plane ordering of children would be additional structure not present in the question.