A fundamental problem in biological classification is the reconstruction of phylogenetic trees for a set X of species from a collection of either subtrees or qualitative characters. This task is equivalent to tree reconstruction from a set of partial X-splits (bipartitions of subsets of X). In this paper, we define and analyse a "closure" operation for partial X-splits that was informally proposed by Meacham [5]. In particular, we establish a sufficient condition for such an operation to reconstruct a tree when there is essentially only one tree that displays the partial X-splits. This result exploits a recent combinatorial result from [2].