States are sampled according to multinomial weights within each subtree, but the subtrees themselves are not sampled uniformly but rather biased towards subtrees away from the initial point. The former uses w_{new} / (w_{new} + w_{old}) whereas the latter uses w_{new} / w_{old}.
See Section A.3.2 of https://arxiv.org/abs/1701.02434 for a more in depth discussion and proof of validity.