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ButcherBeta

ButcherBeta[tree]
gives the number of ways of labeling tree with ButcherOrder[tree]-1 distinct labels such that the root is not labeled, but every other vertex is labeled.
ButcherBeta[n, tree]
gives the number of ways of labeling n of the vertices of tree with n distinct labels such that every leaf is labeled and the root is not labeled.