  A rooted tree is called a $k$-ary tree, if all non-leaf vertices have exactly
$k$ children, except possibly one non-leaf vertex has at most $k-1$ children.
Denote by $h(k)$ the minimum integer such that every tournament of order at
least $h(k)$ contains a $k$-ary spanning tree. It is well-known that every
tournament contains a Hamiltonian path, which implies that $h(1)=1$. Lu et al.
[J. Graph Theory {\bf 30}(1999) 167--176] proved the existence of $h(k)$, and
showed that $h(2)=4$ and $h(3)=8$. The exact values of $h(k)$ remain unknown
for $k\geq 4$. A result of Erd\H{o}s on the domination number of tournaments
implies $h(k)=\Omega(k\log k)$. In this paper, we prove that $h(4)=10$ and
$h(5)\geq13$.
