Each vertex consumes a binary sequence

Halving a positive number removes its least significant binary digit. Thus vertex $v$ must participate in exactly $\ell(a_v)$ moves, where $\ell(x)$ is the number of binary digits of $x$.

Write those digits from least significant to most significant. This sequence records the parity of the vertex before each of its moves. Do not add leading zeros: after the last digit is consumed, the vertex disappears.

Let $L(x,y)$ be the longest common subsequence length of these two digit sequences. If adjacent vertices are selected together several times, the corresponding parities form a common subsequence. Consequently, they can share at most $L(x,y)$ moves.