Complexity issues for Preorders on finite labeled forests

We prove that three preorders on the finite k-labeled forests are polynomial time computable. Together with an earlier result of the first author, this implies polynomial-time computability for an important initial segment of the corresponding degrees of discontinuity of k- partitions on the Baire space. Furthermore, we show that on ω-labeled forests the first of these three preorders is polynomial time computable as well while the other two preorders are NP-complete.