def to_poset(elt, root_to_leaf = False):
    if elt in LabelledOrderedTrees():
        relabel = False
    else:
        elt = elt.canonical_labelling()
        relabel = True
    relations = []
    elements = [elt.label()]
    roots = [elt]
    while len(roots)!=0:
        node = roots.pop()
        for child in node:
            elements.append(child.label())
            relations.append((node.label(),child.label()) if root_to_leaf else (child.label(),node.label()))
            roots.append(child)
    from sage.combinat.posets.posets import Poset
    p = Poset([elements, relations])
    if relabel:
        p = p.canonical_label()
    return p

@cached_function
def preimages(level):
    result = dict()
    for el in OrderedTrees(level):
        image = to_poset(el)._hasse_diagram.canonical_label().copy(immutable=True)
        result[image] = result.get(image, 0) + 1
    return result

def statistic(x):
    return preimages(x.cardinality()).get(x._hasse_diagram.canonical_label().copy(immutable=True), 0)