If you look at this as a series of consecutive operations, every odd operation grows by a little over three, and is guaranteed to be followed by an even operation. Even operations shrink by a little more than three, so all that's required is to compare the cardinality of those two offsets to see whether continued operations are growing or shrinking. In the case of odd operations, as the infinite series of operations continues, the offset gets smaller and smaller. In contrast, the size of the offset for even operations doesn't change as the series of operations continues, which should mean that numbers shrink more than the grow, just ever so slightly.
(To understand the even operations, you have to look at expected values. 1/2 of all even operations will produce an odd operation after dividing by two, 1/4 after dividing by four, 1/8 after dividing by eight. The infinite sum (1/2)^2n converges to 1/3, with an actual value of 1/3 - (1/3)*(1/2)^2n.