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

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