Absolutely symmetric trees and complexity of natural number

TL;DR

This paper explores the relationship between absolutely symmetric rooted trees and the complexity of natural numbers, establishing formulas and properties that connect tree structures with number complexity.

Contribution

It introduces a novel concept linking symmetric trees to natural number complexity and derives formulas for calculating and comparing these complexities.

Findings

Difference in complexities of consecutive numbers equals 1 iff the number is simple.

A recurrent ratio for the complexity of natural numbers is established.

Formulas for calculating differences in complexities are provided.

Abstract

We consider the rooted trees which not have isomorphic representation and introduce a conception of complexity a natural number also. The connection between quantity such trees with $n$ edges and a complexity of natural number $n$ is established. The recurrent ratio for complexity of a natural number is founded. An expression for calculation of difference complexities of two adjacent natural numbers is constructed. It is proved that this difference equal 1 if and only if a natural number is simple. From proved theorems it follows corollaries.