Basic Examples

Get the additional trees that are necessary for a fourth-order method assuming that the quadrature conditions through order 4 and the row and column simplifying assumptions of order 1 hold:
[◼]
ButcherTreeSimplify
[4,1,1]
In[]:=
{{f.[f.f.[f.]]}}
Out[]=

Properties and Relations

The non-reduced trees for any order can be obtained using ButcherTrees:
ResourceFunction["ButcherTrees"][4]
In[]:=
{f.},{f.[f.]},f.[f.[f.]],f.
2
f.
,f.[f.[f.[f.]]],f.f.
2
f.
,f.[f.f.[f.]],f.
3
f.

Out[]=