Non-isomorphic graphs with the same subtree polynomial

Let $n$ be a positive integer, and let the subtree polynomial of a graph $G$ be

$$s_G(x)=\sum_{k=0}^{n-1}a_kx^k,$$

where $a_k$ counts the subtrees with $k$ edges. Subtree-polynomial multiplicity conjecture. For any $n$, one can construct $n$ pairwise non-isomorphic graphs that all share the same subtree polynomial. The paper motivates this by pigeonhole considerations, since there should be many more graphs on $n$ vertices than possible subtree polynomials, but gives no resolution in the supplied text.