  In this paper I present a new method of studying the densities of the Collatz
trajectories generated by a set $S \subset \mathbb{N}$. This method is used to
furnish an alternative proof that $d(\{y \in \mathbb{N} : \exists k \text{
where } T^k(y) < cy\}) = 1$ for all $c > 0$. Finally, I briefly discuss how the
ideas presented in this paper could be used to improve the result that $d(\{y
\in \mathbb{N} : \exists k \text{ where } T^k(y) < y^c\}) = 1$ for $c > \log_4
3$.
