Runge-Kutta methods are some of the most widely used numerical integrators for approximat-
ing the solution of an ordinary di erential equation (ODE). These methods form a subset of a
larger class of numerical integrators called B-series methods. B-Series methods are expressed in
terms of rooted trees, a type of combinatorial graph, which are related to the vector eld of the
ODE that is to be solved. Therefore, the conditions for B-series methods to preserve important
properties of the solution of an ODE, such as symplecticity and energy-preservation, may be ex-
pressed in terms of rooted trees. Certain linear combinations of rooted trees give conditions for a
B-series to be Energy-preserving while other linear combinations give conditions for a B-series to
be Hamiltonian. B-series methods may be conjugate (by another B-series) to an Energy-preserving
or an Hamiltonian B-series. Such B-series methods are called conjugate-to-Energy preserving and
conjugate-to-Hamiltonian, respectivley. The conditions for a B-series to be conjugate-to-Energy
preserving or conjugate-to-Hamiltonian may also be expressed in terms of rooted trees.
The rooted trees form a vector space over the Real numbers. This thesis explores the algebraic
structure of this vector space and its natural energy-preserving, Hamiltonian, conjugate-to-Energy
preserving and conjugate-to-Hamitlonian subspaces and dual subspaces.
The rst part of this thesis reviews important concepts of numerical integrators and introduces
the general Runge-Kutta methods. B-series methods, along with rooted trees, are then introduced
in the context of Runge-Kutta methods. The theory of rooted trees is developed and the conditions
for a B-series to be Hamiltonian or have rst integral are given and discussed. In the nal chapter
we interpret the conditions in the context of vector spaces and explore the algebraic structure of,
and the relationships between, the natural vector subspaces and dual spaces.