  The Collatz graph is a directed graph with natural number nodes and where
there is an edge from node $x$ to node $T(x)=T_0(x)=x/2$ if $x$ is even, or to
node $T(x)=T_1(x)=\frac{3x+1}{2}$ if $x$ is odd. Studying the Collatz graph in
binary reveals complex message passing behaviors based on carry propagation
which seem to capture the essential dynamics and complexity of the Collatz
process. We study the set $\mathcal{E} \text{Pred}_k(x)$ that contains the
binary expression of any ancestor $y$ that reaches $x$ with a limited budget of
$k$ applications of $T_1$. The set $\mathcal{E} \text{Pred}_k(x)$ is known to
be regular, Shallit and Wilson [EATCS 1992].
  In this paper, we find that the geometry of the Collatz graph naturally leads
to the construction of a regular expression, $\texttt{reg}_k(x)$, which defines
$\mathcal{E} \text{Pred}_k(x)$. Our construction, is exponential in $k$ which
improves upon the doubly exponentially construction of Shallit and Wilson.
Furthermore, our result generalises Colussi's work on the $x = 1$ case [TCS
2011] to any natural number $x$, and gives mathematical and algorithmic tools
for further exploration of the Collatz graph in binary.
