  Let $G$ be a connected graph and let $k$ be a positive integer. Let $T$ be a
spanning tree of $G$. The leaf degree of a vertex $v\in V(T)$ is defined as the
number of leaves adjacent to $v$ in $T$. The leaf degree of $T$ is the maximum
leaf degree among all the vertices of $T$. Let $A(G)$ be the adjacency matrix
of $G$ and $D(G)$ be the diagonal degree matrix of $G$. Let $Q(G)=D(G)+A(G)$ be
the signless Laplacian matrix of $G$. The largest eigenvalue of $Q(G)$, denoted
by $q(G)$, is called the signless Laplacian spectral radius of $G$. In this
paper, we investigate the connection between the spanning tree and the signless
Laplacian spectral radius of $G$, and put forward a sufficient condition based
upon the signless Laplacian spectral radius to guarantee that a graph $G$
contains a spanning tree with leaf degree at most $k$. Finally, we construct
some extremal graphs to claim all the bounds obtained in this paper are sharp.
