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The conditions for a Runge--Kutta method to be of order $p$ with $p\ge 5$ fora scalar non-autonomous problem are a proper subset of the order conditions fora vector problem. Nevertheless, Runge--Kutta methods that were derivedhistorically only for scalar problems happened to be of the same order forvector problems. We relate the order conditions for scalar problems tofactorisations of the Runge--Kutta trees into "atomic stumps" and enumeratethose conditions up to $p=20$. Using a special search procedure overunsatisfied order conditions, new Runge--Kutta methods of "ambiguous orders"five and six are constructed. These are used to verify the validity of theresults.

Isomeric trees and the order of Runge--Kutta methods