I discuss the emergence of logarithmic CFT content associated to SLE(8) and non-local observables in the planar uniform spanning tree (UST) model, constructed via scaling limits of Peano curves and their crossing probabilities. In particular, with explicit correlation functions and their fusion thus obtained, one sees that any CFT describing the geometry of UST must be non-unitary (thus not reflection positive). This is of course no surprise - we give a systematic construction directly from the lattice model via its scaling limit, together with immediate relation to SLE(8). Joint work with Mingchang Liu and Hao Wu.