Distant parents in complete binary trees

TL;DR

This paper introduces the concept of distant parents in complete binary trees, revealing new links to dyadic rationals and offering fresh insights into continued fractions as optimal rational approximations.

Contribution

It defines distant parents in complete binary trees and explores their connections to dyadic rationals and continued fractions, providing new theoretical insights.

Findings

Distant parents in binary trees relate to dyadic rationals.

New perspective on continued fractions as best rational approximations.

Establishment of novel connections between tree structures and number theory.

Abstract

There is a unique path from the root of a tree to any other vertex. Every vertex, except the root, has a parent: the adjoining vertex on this unique path. This is the conventional definition of the parent vertex. For complete binary trees, however, we show that it is useful to define another parent vertex, called a \emph{distant parent}. The study of distant parents leads to novel connections with dyadic rational numbers. Moreover, we apply the concepts of close and distant parent vertices to deduce an apparently new sense in which continued fractions are `best' rational approximations.