On tree hook length formulae, Feynman rules and B-series

This thesis relates similar ideas from enumerative combinatorics, Hopf algebraic quantum field theory and differential analysis. Hook length formulae, from enumerative combinatorics, are equations that can lead to bijections between tree classes and other combinatorial classes. Feynman rules are maps used in quantum field theory to generate integrals from particle interaction diagrams. Here we consider Feynman rules from the Hopf algebra perspective. B-series are powers series that sum over trees and are used in differential analysis to analyze Runge-Kutta methods. The aim of this thesis is to bring together the ideas of the three communities. We show how to use differential equations to obtain new hook length formulae. Some of these new hook length formulae result in new combinatorial bijections. We use hook length formulae to express differential equations combinatorially. We also provide a generalization to hook length. Finally we include a catalogue of known hook length formulae.