An extremal theorem for non-isomorphic spanning trees

For a graph $G$, let $τ_{\mathrm{iso}}(G)$ denote the number of isomorphism classes of its spanning trees. For every fixed $d\ge3$ and all sufficiently large $n$, we prove that every connected $n$-vertex graph $G$ with $δ(G)\ge d$ satisfies \[τ_{\mathrm{iso}}(G)\ge τ_{\mathrm{iso}}(K_{d,n-d})=A_dn^{