### Filtering certain posets

I want to filter the set of all connected posets with n points using sage.
I want to filter out the posets whose underlying undirected graph of its Hasse diagram is not a tree (equivalently, the incidence algebra is not hereditary) and such that there is not an element x in the poset P such that any other element of P is comparable to y.
Is there an easy method to do this with sage?