If we choose an incoming hyperedge according to the probability P t (τ(z)) of the resulting tree, holding all other variable assignments fixed, we see that, because P t is uniform, we will choose with uniform probability of 1/m among the m incoming hyperedges at each node.
we will choose among the two incoming hyperedges at the root (node 1) with equal probability, meaning that, over the long run, the sampler will spend half its time in the state for the single tree corresponding to nodes 2 and 3, and only one eighth of its time in each of the four other possible trees.