University of Cambridge

Logic
&
Semantics

Reconstruction of X-trees from subtrees

By Mike Steel (14th August 1998)
Biomathematics Research Centre, University of Canterbury, Christchurch, New Zealand

An X-tree is a finite tree T=(V,E), together with a map tex2html_wrap_inline21 for which

displaymath23

Such trees are equivalent to ``pairwise compatible" system of bipartitions of X, and are widely used in classification, particularly in biology, but also in linguistics, philology etc.

In such applications we are often given a list tex2html_wrap_inline25 where tex2html_wrap_inline27 is a tex2html_wrap_inline29 -tree, for a (small) subset tex2html_wrap_inline29 of X, and we wish to determine if these trees can be consistently combined into one (or more) unknown ``parent" X-tree(s). This tree reconstruction problem is closely related to another which takes as its input a collection of partitions of subsets of X.

We will discuss some of the resulting combinatorial and computational features of this problem. In particular we consider the class of all possible ``inference rules" for enlarging any consistent collection of trees. We show that (as conjectured) this class is infinite, but under certain restrictions a small finite subset always suffices to reconstruct a tree. We also consider the question of whether or not this system of inference rules suffices to determine the consistency of tex2html_wrap_inline39 .

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