Patching Up X-trees

A fundamental problem in many areas of classification, and particularly in biology, is the reconstruction of a leaf-labeled tree from just a subset of its induced subtrees. Without loss of generality, we may assume that these induced subtrees all have precisely four leaves. Of particular interest is the question of determining whether a collection of quartet subtrees uniquely defines a parent tree. Here, we solve this question in the case where the collection of quartet trees is of minimal size, by studying encodings of binary trees by such quartet trees. We obtain a characterization of minimal encodings that exploits an underlying “patchwork” structure. As we will show elsewhere, this allows one to obtain a polynomial time algorithm for certain instances of the problem of reconstructing trees from subtrees.