The Recursive Stopping Time Structure of the $3x+1$ Function

Abstract: The $3x + 1$ problem concerns iteration of the map $T:\mathbb{Z}\rightarrow\mathbb{Z}$ given by \begin{align*} T(x)=\left{\begin{array}{lcr}\;\;\;\;\displaystyle{\frac{x}{2}} & \mbox{if $x\equiv 0\ (\text{mod}\ 2)$},\ \\displaystyle{\frac{3x+1}{2}} & \mbox{if $x\equiv 1\ (\text{mod}\ 2)$}.\end{array}\right. \end{align*} The $3x+1$ Conjecture states that every $x\geq1$ has some iterate $T^s(x)=1$. The least $s\in\mathbb{N}$ such that $T^s(x)<x$ is called the stopping time of $x$. It is shown that the congruence classes $(\text{mod}\ 2^k)$ of the integers having finite stopping time are given by a recursive algorithm producing a directed rooted tree.