  We introduce a full binary directed tree structure to represent the set of
natural numbers, further categorizing them into three distinct subsets: pure
odd numbers, pure even numbers, and mixed numbers. We adopt a binary string
representation for natural numbers and elaborate on the composite methodology
encompassing odd- and even-number functions. Our analysis focuses on examining
the iteration sequence (or composition) of the Collatz function and its reduced
variant, which serves as an analog to the inverse function, to scrutinize the
validity of the Collatz conjecture. To substantiate this conjecture, we
incorporate binary strings into an algebraic formula that captures the essence
of the Collatz sequence. By this means, we transform discrete powers of 2 into
continuous counterparts, ultimately culminating in the smallest natural number,
1. Consequently, the sequence generated through infinite iterations of the
Collatz function emerges as an eventually periodic sequence, thereby validating
an enduring 87-year-old conjecture.
