Algorithm for finding structures and obstructions of tree ideals

Let I be any topological minor closed class of trees (a tree ideal). A classical theorem of Kruskal [Well-quasi-ordering, the Tree Theorem, and Vazsonyi's conjecture, Trans. Am. Math. Soc. 95 (1960) 210-223] states that the set O(I) of minimal non-members of I is finite. On the other hand, a finite structural description S(I) is developed by Robertson, et al. [Structural descriptions of lower ideals of trees, Contemp. Math. 147 (1993) 525-538]. Given either of the two finite characterizations of I, we present an algorithm that computes the other.