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Numerical Differential Equation Analysis Package Symbol

ButcherBetaBar

ButcherBetaBar[tree]
gives the number of ways of labeling tree with ButcherOrder[tree] distinct labels such that every vertex is labeled.

ButcherBetaBar[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.

To use ButcherBetaBar, you first need to load the Numerical Differential Equation Analysis Package using Needs["NumericalDifferentialEquationAnalysis`"].