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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?

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 no 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?