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Abstract |
We introduce a new arithmetic for non-empty rooted unordered trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication, and stretch, and prove their properties. Using these operations all trees can be generated from a starting tree of one vertex. We show how a given tree can be obtained as the sum or as the product of two trees, and define 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 introduce tree equations whose coefficients are integers and whose unknowns are trees. We show how to solve some tree equations as an introduction to the field, and suggest more advanced examples. Finally we briefly discuss how our arithmetic might be useful in different applications. To the best of our knowledge our proposal is new and may be susceptible of variations and improvements. |
