Let $G$ be a connected simple graph. For any spanning tree $T$ of $G$, let $l(T)$ be the number of leaves of the graph $T$. Consider $\ell=\min_Tl(T)$, can I find a spanning tree $T$ with $l(T)=\ell$, such that the set of leaves $A$ of $T$ is very close to an independent set.

For example, I guess that there exists a vertex $a\in A$ such that $A\setminus\{a\}$ is an independent set in the graph $G$.

My idea is that maybe we can justifying the tree $T$ step by step to get a extreme tree with some properties.

Is there any results or references for this question?