  In the \emph{budgeted rooted node-weighted Steiner tree} problem, we are
given a graph $G$ with $n$ nodes, a predefined node $r$, two weights associated
to each node modelling costs and prizes. The aim is to find a tree in $G$
rooted at $r$ such that the total cost of its nodes is at most a given budget
$B$ and the total prize is maximized. In the \emph{quota rooted node-weighted
Steiner tree} problem, we are given a real-valued quota $Q$, instead of the
budget, and we aim at minimizing the cost of a tree rooted at $r$ whose overall
prize is at least $Q$.
  For the case of directed graphs with additive prize function, we develop a
technique resorting on a standard flow-based linear programming relaxation to
compute a tree with good trade-off between prize and cost, which allows us to
provide very simple polynomial time approximation algorithms for both the
budgeted and the quota problems. For the \emph{budgeted} problem, our algorithm
achieves a bicriteria $(1+\epsilon,
O(\frac{1}{\epsilon^2}n^{2/3}\ln{n}))$-approximation, for any $\epsilon \in (0,
1]$. For the \emph{quota} problem, our algorithm guarantees a bicriteria
approximation factor of $(2, O(n^{2/3}\ln{n}))$. Next, by using the flow-based
LP, we provide a surprisingly simple polynomial time $O((1+\epsilon)\sqrt{n}
\ln {n})$-approximation algorithm for the node-weighted version of the directed
Steiner tree problem, for any $\epsilon>0$.
  For the case of undirected graphs with monotone submodular prize functions
over subsets of nodes, we provide a polynomial time
$O(\frac{1}{\epsilon^3}\sqrt{n}\log{n})$-approximation algorithm for the
budgeted problem that violates the budget constraint by a factor of at most
$1+\epsilon$, for any $\epsilon \in (0, 1]$. Our technique allows us to provide
a good approximation also for the quota problem.
