Let $k$, $d$ be a positive integer, $G$ be a connected graph of order $n$, $T$ be a tree. The leaf distance of a tree is defined as the minimum distance between any two leaves. For $v\in V(T)$, the leaf degree of $v$ in $T$ is the number of leaves adjacent to $v$, and the leaf degree of $T$ is defined as maximum leaf degree among the vertices of $T$. In this paper, motivated by the conjecture proposed by Kaneko (2001) and its subsequent partial confirmation by Erbes, Molla, Mousley and Santana (2017), we obtain lower bounds in terms of the size and the adjacent spectral radius to guarantee that $G$ contains a spanning tree with leaf distance at least $d$, where $4\leq d \leq n-1$. Furthermore, we obtain some tight conditions in $G$ for its size and spectral radius to ensure that $G$ has a spanning tree with leaf degree at most $k$, which improves the result of Ao, Liu, Yuan, Ng and Cheng (2023).