Rooted trees, non-rooted trees and hamiltonian B-series

TL;DR

This paper investigates the mathematical relationship between rooted and non-rooted trees, providing a new algebraic structure for non-rooted trees that substitutes the Lie bracket using antisymmetrization of the pre-Lie product.

Contribution

It introduces a novel algebraic framework connecting rooted and non-rooted trees, including a substitute for the Lie bracket in non-rooted trees.

Findings

Established a relationship between rooted and non-rooted trees.

Provided a substitute for the Lie bracket in non-rooted trees.

Linked the structures to hamiltonian B-series.

Abstract

We explore the relationship between (non-planar) rooted trees and free trees, i.e. without root. We give in particular, for non-rooted trees, a substitute for the Lie bracket given by the antisymmetrization of the pre-Lie product.