An Arithmetic for Rooted Trees

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 prove that factorization is unique. We then define negative trees and suggest dealing with tree equations, giving some preliminary examples. Finally we comment on how our arithmetic might be useful, and discuss preceding studies that have some relations with ours. The parts of this work that do not concur to an immediate illustration of our proposal, including formal proofs, are reported in the Appendix. To the best of our knowledge our proposal is completely new and can be largely modified in cooperation with the readers. To the ones of his age the author suggests that “many roads must be walked down before we call it a theory”. 1998 ACM Subject Classification E.1 Data Structures, G.2.0 [Discrete Mathematics] General, G.2.2 [Discrete Mathematics] Graph Theory