Output polynomial enumeration of all fixed-cardinality ideals of a   poset, respectively all fixed-cardinality subtrees of a tree
  The N cardinality k ideals of any w-element poset (w, k variable) can be enumerated in time O(Nw^3). The corresponding bound for k-element subtrees of a w-element tree is O(Nw^5). An algorithm is described that by the use of wildcards displays all order ideals of a poset in a compact manner, i.e. not one by one. 
Marcel Wild
math.CO