  We consider an incremental variant of the rooted prize-collecting
Steiner-tree problem with a growing budget constraint. While no incremental
solution exists that simultaneously approximates the optimum for all budgets,
we show that a bicriterial $(\alpha,\mu)$-approximation is possible, i.e., a
solution that with budget $B+\alpha$ for all $B \in \mathbb{R}_{\geq 0}$ is a
multiplicative $\mu$-approximation compared to the optimum solution with budget
$B$. For the case that the underlying graph is a tree, we present a
polynomial-time density-greedy algorithm that computes a
$(\chi,1)$-approximation, where $\chi$ denotes the eccentricity of the root
vertex in the underlying graph, and show that this is best possible. An
adaptation of the density-greedy algorithm for general graphs is
$(\gamma,2)$-competitive where $\gamma$ is the maximal length of a
vertex-disjoint path starting in the root. While this algorithm does not run in
polynomial time, it can be adapted to a $(\gamma,3)$-competitive algorithm that
runs in polynomial time. We further devise a capacity-scaling algorithm that
guarantees a $(3\chi,8)$-approximation and, more generally, a
$\smash{\bigl((4\ell - 1)\chi, \frac{2^{\ell +
2}}{2^{\ell}-1}\bigr)}$-approximation for every fixed $\ell \in \mathbb{N}$.
