The arithmetic of trees

TL;DR

This paper explores an extension of natural number arithmetic to planar binary trees, creating a non-commutative arithmetic system called arithmetree, and investigates properties of prime trees within this framework.

Contribution

It introduces and analyzes the arithmetree, a novel non-commutative arithmetic on trees, expanding the understanding of tree-based algebraic structures.

Findings

Definition and properties of arithmetree

Characterization of prime trees in this system

Extension of natural number arithmetic to trees

Abstract

The arithmetic of the natural numbers can be extended to arithmetic operations on planar binary trees. This gives rise to a non-commutative arithmetic theory. In this exposition, we describe this arithmetree, first defined by Loday, and investigate prime trees.