Anyon theories are required to be associative, i.e. when fusing three anyons with labels $a,b,c$, we have $$(a\times b) \times c = a\times (b \times c)$$ This associativity is extended to the fusion and splitting spaces. Vectors in these spaces are represented by fusion trees as in the image. The image suggests that there are precisely two ways of fusing $a,b,c$ to $d$
My question is simple, what about first fusing $a$ with $c$ and then with $b$? Should this not be another possibility that is not covered by the diagrams. Indeed, in his paper https://arxiv.org/abs/cond-mat/0506438 Kitaev explicitly points out that the anyon theory is established on a line and that the order of anyons on that line matters. Still the theory is claimed to describe particles in 2D. I cannot see how this fits, as in 2D, the anyons need not be arranged on a line but can be literally anywhere.