How to detect "tree-able" set-families?

A set-family (a set of sets of elements) is called tree-able 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 set-family is treeable (and construct a tree)?