A setfamily (a set of sets of elements) is called treeable if the elements can be arranged on a directed tree such that each set in the family corresponds to a path from the root to a leaf. For example, the following family is treeable:
{ {1, 2, 6}, {1, 3, 4, 5, 6} }
A possible tree is:
1 > 6 > 2

+ > 3 > 4 > 5
However, the following family is not treeable:
{ {1, 2}, {2, 3}, {3, 1} }
Is it possible to decide in polynomial time whether a given setfamily is treeable (and construct a tree)?