  A $k$-tree is a spanning tree in which every vertex has degree at most $k$.
In this paper, we provide a sufficient condition for the existence of a
$k$-tree in a connected graph with fixed order in terms of the adjacency
spectral radius and the signless Laplacian spectral radius, respectively. Also,
we give a similar condition for the existence of a perfect matching in a
balanced bipartite graph with fixed order and minimum degree.
