The path sets of weighted partially labelled trees

Massey University Palmerston North New Zealand In the construction of evolutionary trees for a set of n species, we are sometimes given a set of measures of differences between each pair of species. The underlying assump:ion is that these Jifferences are the result of a stochastic process of changes occurring on the edges of the historical tree T which incorporates the evolutionary relationships of the species. We quantify the "changes" on the edges of T as additive edge weights where the sum of edge weightson the path between two species is their difference value. A fundamental problem of biology is to determine T and its edge weight funcT;on w, given only the set of difference values between them. For real data an exact fit cannot be expected. A number of algorithms exist, of varying complexity, that estimate a tree which purports to represent T. Although the biological model implies that the edge weights are non-negative, a theoretical analysis of the effect of data errors requires the consideration of negative weights. In this paper we establish the precise conditions under which the weighted tree (T, w) can be uniquely determined, that is the conditions for the mapping from (T, w) to the set of differences to be one to one. This is done in the general case where the differences can be negative, extending the classical result for positive differences. We prove that inversion is possible if and only "if the weight of each edge is nonzero and every vertex of degree 1 and 2 of the tree represents one of the species being considered. The proof of this result uses a natural application of a Hadamard matrix and provides us with an algorithm for reconstructing (T, w) from the set of difference values. However this algorithm is not practical for real data, but a closely related practical algorithm exists when the edge weights are positive. Introduction Let T = (V, E) be a tree with edge set E and vertex set V. We say T is an (edge) weighted tree <=:> ~ w: E --7 R. We say T is partially (vertex) labelled by the set L<=:> 31: L --7 V. Let (T, w) be a weighted t~ee partially labelled byN = {I, 2, ... , n}. For i, j E N, let I1ij be the (i, j)th path, the set vf edges of E connecting vertices lei) and t(j). For i, j E N, let Dij = L,w(e), Dij 0 and D(T, w) = [Dij I i, j E N] be e E I1ij the difference matrix of (T, w). The major result of this paper is: Theorem Given the difference matrix D(T, w) we can reconstruct (T, w) <=> 1. Vv E V, d(v) < 3 => V E t(N); and 2. wee) :f0, Ve E E. Note The corresponding result for positive weights was established by Buneman [1971]. This extended result can also be shown to follow from the linear independence of split metrics independently developed by Bandelt and Dress [1990]. Example 1