The normal binary approach is used simply because it's easy and symmetrical. The main goal is to make the transmission line splits keep the same net impedance at each junction by parallelism, e.g. a 50 Ohm feed line splits to two 100 Ohm which each split to two 200 Ohms lines and so on. There's still a potential discontinuity at the junction as the transition isn't perfectly uniform on a microscopic scale and the current has to turn a corner, etc. Thus, making the two transmission lines split opposite each other helps to minimize any imbalance in the feed transition.

So, on the first rule of maintaining impedance, there's no reason you couldn't split three ways (e.g. 50 Ohm to 3x 150 Ohm) but the symmetry of the transition now becomes a bit more challenging. There will always be a "center" transmission line that would invariably see a slightly different wave propagation relative to the two on either side. Any alternative there would be to remove the planar transition and instead feed the splitter from a via (e.g. feed on one side of the ground plane and tri-lateral symmetric transmission lines on the other) which would solve the symmetry problem at the split, but at the cost of a lot of mismatch error at the via. On top of that, regardless of which approach you use, you have a distribution problem where if you're lining the antenna elements up in anything other than triangular symmetry, you have to use meander lines to keep the electrical length of the split feeds the same.

The alternative of course would be to go back to the 2x2 split configuration and just terminate the fourth line you don't need. That of course loses a quarter of the power you're applying, so that's not an ideal solution either.

Hopefully that helps.

Michael