Most computational models of dependency syntax consist of distributions over spanning trees. However, the majority of dependency treebanks require that every valid dependency tree has a single edge coming out of the ROOT node, a constraint that is not part of the definition of spanning trees. For this reason all standard inference algorithms for spanning trees are sub-optimal for inference over dependency trees.Zmigrod et al (2021) proposed algorithms for sampling with and without replacement from the dependency tree distribution that incorporate the single-root constraint. In this paper we show that their fastest algorithm for sampling with replacement, Wilson-RC, is in fact producing biased samples and we provide two alternatives that are unbiased. Additionally, we propose two algorithms (one incremental, one parallel) that reduce the asymptotic runtime of algorithm for sampling k trees without replacement to O(kn^3). These algorithms are both asymptotically and practically more efficient.