<i>T</i>₃ Tree and Its Traits in Understanding Integers

Abstract

Valuated binary tree is emerging its magical characteristics in study of integers. As the father tree of all valuated binary subtrees, T 3 tree plays a very important role in studying the odd integers for its owning both the common properties of a general valuated binary tree and its own traits of covering all the odd integers bigger than 1. The article investigates the properties of the T 3 tree as well as the multiplications on the tree. It exhibits the distribution of a node's multiples, the distribution of the product of two nodes and the distribution of the square of a node. Some other contents such as fundamental properties of the T 3 tree and path connecting a product to its divisor-nodes are also investigated with detail mathematical proofs. The theorems and corollaries proved in the article lay a foundation for future work.