Arithmetic for Rooted Trees

TL;DR

This paper introduces a novel arithmetic system for rooted unordered trees, defining operations like addition and multiplication, and explores their properties, primality, and factorization, with potential applications in tree equations.

Contribution

It presents the first comprehensive arithmetic framework for rooted trees, including prime trees, primality testing, and unique factorization, which are new contributions in this domain.

Findings

Defined tree addition, multiplication, and stretch operations

Proved primality testing in polynomial time

Established unique factorization of trees

Abstract

We propose a new arithmetic for non-empty rooted unordered trees simply called trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication and stretch, prove their properties, and show that all trees can be generated from a starting tree of one vertex. We then show how a given tree can be obtained as the sum or product of two trees, thus defining prime trees with respect to addition and multiplication. In both cases we show how primality can be decided in time polynomial in the number of vertices and we prove that factorization is unique. We then define negative trees and suggest dealing with tree equations, giving some preliminary results. Finally we comment on how our arithmetic might be useful, and discuss preceding studies that have some relations with our. To the best of our knowledge our approach and results are completely new aside for an earlier version of this work submitte as an arXiv manuscript.