Backwards iteration of the $3x+1$ function starting from a fixed
integer a produces a tree of preimages of a. Let $\T_{k} (a)$
denote this tree grown to depth k, and let $\T^*_{k}(a)$ denote the
pruned tree resulting from the removal of all nodes $n \equiv 0
\pmod{3}$. We previously computed the maximal and minimal number of leaves in
$\T^*_{k} (a)$ for all $a \not\equiv 0 \pmod{3}$ and all $k\le30$. Here we
compare these data with predictions made using branching process
models designed to imitate the growth of $3x+1$ trees, developed
in [Lagarias and Weiss 1992]. We derive rigorous results for the branching process
models. The range of variation exhibited by the $3x+1$ trees appears
significantly narrower than that of the branching process models. We
also study the variation in expected leaf-counts associated to the
congruence class of $a \pmod{3^j}$. This variation, when properly
normalized, converges almost everywhere as $j \to \infty$ to a
limit function on the invertible 3-adic integers.