  We consider a variant of the prize collecting Steiner tree problem in which
we are given a \emph{directed graph} $D=(V,A)$, a monotone submodular prize
function $p:2^V \rightarrow \mathbb{R}^+ \cup \{0\}$, a cost function $c:V
\rightarrow \mathbb{Z}^{+}$, a root vertex $r \in V$, and a budget $B$. The aim
is to find an out-subtree $T$ of $D$ rooted at $r$ that costs at most $B$ and
maximizes the prize function. We call this problem \emph{Directed Rooted
Submodular Tree} (\textbf{DRSO}).
  Very recently, Ghuge and Nagarajan [SODA\ 2020] gave an optimal
quasi-polynomial-time $O\left(\frac{\log n'}{\log \log
n'}\right)$-approximation algorithm, where $n'$ is the number of vertices in an
optimal solution, for the case in which the costs are associated to the edges.
  In this paper, we give a polynomial-time algorithm for \textbf{DRSO} that
guarantees an approximation factor of $O(\sqrt{B}/\epsilon^3)$ at the cost of a
budget violation of a factor $1+\epsilon$, for any $\epsilon \in (0,1]$. The
same result holds for the edge-cost case, to the best of our knowledge this is
the first polynomial-time approximation algorithm for this case. We further
show that the unrooted version of \textbf{DRSO} can be approximated to a factor
of $O(\sqrt{B})$ without budget violation, which is an improvement over the
factor $O(\Delta \sqrt{B})$ given in~[Kuo et al.\ IEEE/ACM\ Trans.\ Netw.\
2015] for the undirected and unrooted case, where $\Delta$ is the maximum
degree of the graph. Finally, we provide some new/improved approximation bounds
for several related problems, including the additive-prize version of
\textbf{DRSO}, the maximum budgeted connected set cover problem, and the
budgeted sensor cover problem.
