In the maximum-weight degree-bounded forest problem, we are given a graph with nonnegative edge weights and a degree upper bound on each vertex. The objective is to find a maximum-weight forest satisfying the degree bounds. We show that this problem admits a simple $2$-approximation via the greedy algorithm. This already improves over the best-known guarantees obtained via a reduction to weighted $3$-matroid parity, and sets a baseline to improve upon. Our main result is an $\frac{8}{5}$-approximation algorithm for the maximum-weight degree-bounded forest problem. For the special case of uniform degree bounds, our algorithm achieves an approximation ratio of $\frac{3}{2}$. It computes two sets of edges, each of which we then restrict to a feasible solution in an optimal way, and finally returns the better of the two resulting solutions. To obtain the initial two edge sets, we first relax the degree bounds and leverage an algorithm by Lau and Singh [JACM'15] for degree-bounded spanning trees, and second we relax the forest constraints and employ an algorithm for maximum-weight $b$-matchings.

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