An extremal theorem for non-isomorphic spanning trees

For a graph $G$, let ${\tau }_{\mathrm{i}\mathrm{s}\mathrm{o}}(G)$ denote the number of isomorphism classes of its spanning trees. For every fixed $d\ge 3$ and all sufficiently large $n$, we prove that every connected $n$-vertex graph $G$ with $\delta (G)\ge d$ satisfies $${\tau }_{\mathrm{i}\mathrm{s}\mathrm{o}}(G)\ge {\tau }_{\mathrm{i}\mathrm{s}\mathrm{o}}(K_{d,n-d})=A_d{n}^{d-1}+O_d(n^{d-2}),$$ for an explicit constant $A_d>0$, and $K_{d,n-d}$ is the unique minimizer. This confirms a conjecture of Bitonti, Michel and Scott and extends it to every $d\ge 3$. We also show that any such graph with $O(n^{d-1})$ spanning-tree types has all but a bounded number of vertices with the same $d$ neighbours.