The reconstruction of evolutionary trees from data sets on overlapping sets of species is a central problem in phylogenetics. Provided that the tree reconstructed for each subset of species is rooted and that these trees fit together consistently, the space of all parent trees that display' these trees was recently shown to satisfy the following property: there exists a path from any one parent tree to any other parent tree by a sequence of local rearrangements (nearest neighbour interchanges) so that each intermediate tree also lies in this same tree space. However, the proof of this result uses a non-constructive argument. In this thesis we describe a specific, polynomial-time procedure for navigating from any given parent tree to another while remaining in this tree space. We then investigate a related problem, the conditions under which there is only one parent tree. These results are of particular relevance to the recent study of phylogenetic terraces'.