Abstract
The uniform spanning forest (USF) in Z(d) is the weak limit of random, uniformly chosen, spanning trees in [-n, n]. Pemantle [11] proved that the USF consists a.s. of a single tree if and only if d = 9. More generally, let N(x, y) be the minimum number of edges outside the USF in a path joining x and y in Zd. Then max{N(x,y): x, y epsilon Z(d)} = [(d - 1)/4] a.s. The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.