Annals of Combinatorics 3 (1999) 1-12

Patching Up X-Trees

Sebastian Böcker1, Andreas W.M. Dress1, and Mike A. Steel 2

1GK Struktursprozesse, FSP Mathematisierung, Universität Bielefeld, PF 100 131, 33501 Bielefeld, Germany
{boecker, dress}

2Biomathematics Research Centre, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

Received January 8, 1999

AMS Subject Classification: 05C05, 92D15

Abstract.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 determining whether a collection of quartet subtrees uniquely defines a parent tree. Here, we solve this problem 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.

Keywords: trees, supertrees, tree amalgamation, quartet encodings of trees, hierarchies, patchworks


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