Trees and B-series

  • J. C. ButcherEmail author
Original Paper


The connection between trees and differential equations was pointed out in the classic paper by Cayley (Phil. Mag. 13, 172–176 1857). Trees were also used in the work of Merson (1957), on the order of Runge–Kutta methods. The paper by Hairer and Wanner (Computing 13, 1–15 1974), where the term B-series was introduced, followed papers by the present author (Butcher J. Austral. Math. Soc. 3, 185–201 1963, Math. Comput. 26, 79–106 1972). The present paper will survey the use of trees in the formulation of B-series and illustrate the results by constructing and analysing some examples of general linear methods.


Trees B-series 

Mathematics Subject Classification (2010)



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I am grateful to an anonymous referee for a helpful review of the paper.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand

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