Collatz Trees: A Structural Framework for Understanding the 3x+1 Problem

The Collatz Conjecture remains one of the most enduring unsolved problems in mathematics, despite being based on an extraordinarily simple rule. Given any natural number n, the conjecture posits that repeatedly applying the operation—dividing by 2 if even, or multiplying by 3 and adding 1 if odd—will eventually result in the number 1.This paper develops a structural perspective by proposing the Collatz Tree as a framework to organize and visualize natural numbers. Each branch is the geometric ray {k·2^b} for an odd core k, and the trunk is the ray from 1. We introduce a trunk–branch indexing that bijects N with Z≥0 × Z≥0.Algebraically, we encode Collatz steps as affine maps and prove the absence of nontrivial finite cycles for a three-way map T. Through a bridge theorem, this implies the same for the standard accelerated map A(n) = (3n+1)/2^ν₂(3n+1) on odd integers. Thus, the global Collatz convergence reduces to an independent pillar: the coverage (reachability) of the inverse tree rooted at 1, isolating cycle-freeness from coverage and reducing the conjecture to the remaining reachability problem.This framework provides a unified algebraic and graph-theoretic foundation for future Collatz research.