Partitioning a triangle-free planar graph into a forest and a forest of bounded degree

We prove that every triangle-free planar graph can have its set of vertices partitioned into two sets, one inducing a forest and the other a forest with maximum degree at most 5. We also show that if for some d, there is a triangle-free planar graph that cannot be partitioned into two sets, one inducing a forest and the other a forest with maximum degree at most d, then it is an NP-complete problem to decide if a triangle-free planar graph admits such a partition.