On the random generation of Butcher trees

Abstract: We provide a probabilistic representation of the solutions of ordinary differential equations (ODEs) by random generation of Butcher trees. This approach complements and simplifies a recent probabilistic representation of ODE solutions, by removing the need to generate random branching times. The random sampling of trees allows one to improve numerical accuracy by Monte Carlo iterations whereas the finite order truncation of Butcher series can have higher time complexity, as shown in examples.