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 ...

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. ...

ButcherColumnConditions[p, s] gives the column-simplifying conditions up to and including order p for s stages. ButcherColumnConditions[p] gives the column-simplifying ...

ButcherPhi[tree, s] gives the weight of tree in an s-stage Runge\[Dash]Kutta method. ButcherPhi[tree] gives the weight of tree in stage-independent tensor notation.

ButcherPlot[tree] gives a plot of tree.ButcherPlot[{tree_1, tree_2, ...}] gives an array of plots of tree_1, tree_2, ....

ButcherPrincipalError[p, s] gives the principal error for a method of order p with s stages. ButcherPrincipalError[p] gives the principal error using stage-independent tensor ...

ButcherQuadratureConditions[p, s] gives the quadrature conditions up to and including order p for s stages. ButcherQuadratureConditions[p] gives the quadrature conditions in ...

ButcherRowConditions[p, s] gives the row-simplifying conditions up to and including order p for s stages. ButcherRowConditions[p] gives the row-simplifying conditions in ...

ButcherSimplify is an option to RungeKuttaOrderConditions that specifies whether to apply Butcher's row- and column-simplifying conditions.

ButcherTreeSimplify[p, \[Eta], \[Xi]] gives the set of trees through order p that are not reduced by Butcher's quadrature conditions through order p, row-simplifying ...