  For integer $k\geq2,$ a spanning $k$-ended-tree is a spanning tree with at
most $k$ leaves. Motivated by the closure theorem of Broersma and Tuinstra
[Independence trees and Hamilton cycles, J. Graph Theory 29 (1998) 227--237],
we provide tight spectral conditions to guarantee the existence of a spanning
$k$-ended-tree in a connected graph of order $n$ with extremal graphs being
characterized. Moreover, by adopting Kaneko's theorem [Spanning trees with
constraints on the leaf degree, Discrete Appl. Math. 115 (2001) 73--76], we
also present tight spectral conditions for the existence of a spanning tree
with leaf degree at most $k$ in a connected graph of order $n$ with extremal
graphs being determined, where $k\geq1$ is an integer.
