Can Collatz's problem be used as a pseudo random prime sieve?

Question

If you take the concept of $3x+1$, $\dfrac{x}{2}$ and starting at 2, create a tree. On the left nodes you apply the $3x+1$. On the right nodes, if the parent node is even apply the $\dfrac{x}{2}$. Where $x$ is the value of the parent node. Initial steps show a pseudo-random set of primes and a drastically disproportionate amount of primes to non-primes in contrast to the number line itself. I say pseudo-random, because there is a significant difference of growth between the first path and the next and the last path in each section. Note to exclude 2's right node as it simply creates a loop back to two. Also note that every other level in the tree would be even which means you could skip every other layer in terms of prime checks.