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Two Binary trees of Rational numbers -- the S-tree and the SC-tree

Ziting Wang, Ruijia Guo, Yixin Zhu

Abstract

In this study, we explore a novel approach to demonstrate the countability of rational numbers and illustrate the relationship between the Calkin-Wilf tree and the Stern-Brocot tree in a more intuitive manner. By employing a growth pattern akin to that of the Calkin-Wilf tree, we construct the S-tree and establish a one-to-one correspondence between the vertices of the S-tree and the rational numbers in the interval $(0,1]$ using 0-1 sequences. To broaden the scope of this concept, we further develop the SC-tree, which is proven to encompass all positive rational numbers, with each rational number appearing only once. We also delve into the interplay among these four trees and offer some applications for the newly introduced tree structures.

Two Binary trees of Rational numbers -- the S-tree and the SC-tree

Abstract

In this study, we explore a novel approach to demonstrate the countability of rational numbers and illustrate the relationship between the Calkin-Wilf tree and the Stern-Brocot tree in a more intuitive manner. By employing a growth pattern akin to that of the Calkin-Wilf tree, we construct the S-tree and establish a one-to-one correspondence between the vertices of the S-tree and the rational numbers in the interval using 0-1 sequences. To broaden the scope of this concept, we further develop the SC-tree, which is proven to encompass all positive rational numbers, with each rational number appearing only once. We also delve into the interplay among these four trees and offer some applications for the newly introduced tree structures.
Paper Structure (13 sections, 14 theorems, 17 equations, 15 figures)

This paper contains 13 sections, 14 theorems, 17 equations, 15 figures.

Table of Contents

  1. Introduction
  2. The S-tree and the SC-tree
  3. The S-tree and its properties
  4. Locating vertices of the S-tree
  5. The SC-tree and its properties
  6. Locating vertices of the SC-tree
  7. The linkages of the four trees
  8. The SC-tree and the Stern-Brocot tree
  9. Locating vertices of S-tree in the Calkin-Wilf tree
  10. Application
  11. Fibonacci sequence
  12. Trigonometric function
  13. Conclusion

Key Result

Theorem 2

The S-tree corresponds one-to-one to reduced rational numbers in the interval $(0,1]$.

Figures (15)

  • Figure 1: Illustration of the countability of the positive rationals
  • Figure 2: The Stern–Brocot tree, and the Stern–Brocot sequences of level 1-4
  • Figure 3: The Calkin-Wilf tree of leverl 1-4
  • Figure 4: The tree of product of L and R
  • Figure 5: Growing pattern of the S-tree
  • ...and 10 more figures

Theorems & Definitions (36)

  • Definition 1
  • Theorem 2
  • proof
  • Definition 3
  • Theorem 4
  • proof
  • Example 1
  • Corollary 5
  • Definition 6
  • Definition 7
  • ...and 26 more