The Arithmetic of Even-Odd Trees

Abstract

Even-Odd Trees are a canonical tree-based number representation derived from a bijection between trees defined by the data type equation T = 1+T *T* +T *T* and positive natural numbers seen as iterated applications of o(x) = 2x and i(x) = 2x + 1 starting from 1. This paper introduces purely functional arithmetic algorithms for operations on Even-Odd Trees. While within constant factors from their traditional counterparts for their average case behavior, our algorithms make tractable important computations that are impossible with traditional number representations.