An X-tree is a finite tree T = (V,E), together with a
map φ: X → V for which
degT(v) ≤ 2 ⇒
v ∈&phi(X). 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
T = { T1, ...,
Tk } where Ti is an
Xi-tree, for a (small) subset Xi
of X, and we wish to determine if these trees can be
consistently combined into one or more unknown `parent'
X-trees. 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 T.
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