An inequality for the number of independent sets of matroids with an application to the forest-tree ratio of graphs

Let $M=(E,\mathcal{I})$ be a matroid of rank $r$. Let $\mathcal{I}_k$ be the independent sets of size $k$, and let $I_k=|\mathcal{I}_k|$ and $I=|\mathcal{I}|$. We show that if every set $F\in \mathcal{I}_{r-1}$ is contained in at least $δ$ bases, then $$\ln \left(\frac{I}{I_r}\right)\geqslant \frac{