On dissimilarity vectors of general weighted trees

Abstract

Let T be a (not necessarily positive) weighted tree with n leaves numbered by the set {1, ..., n}. Define the k-weights of the tree D i1,....,i k (T ) as the sum of the lengths of the edges of the minimal subtree connecting i 1 ,....,i k . We will call such numbers "k-weights" of the tree. In this paper, we characterize sets of real numbers indexed by the subsets of any cardinality ≥ 2 of a n-set to be the weights of a tree with n leaves.2000 Mathematical Subject Classification: 05C05, 05C12, 92B05

Treelike Families of Multiweights

“…In [9] and [10], the author gave an inductive characterization of the families of real numbers that are indexed by the subsets of {1, ..., n} of cardinality greater than or equal to 2 and are the families of the multiweights of a tree with n leaves. Let n, k ∈ N with n > k. In [1] we studied the problem of the characterization of the families of positive real numbers, indexed by the k-subsets of an n-set, that are p-treelike in the "border" case k = n − 1.…”

“…We say that a tree P is a pseudostar of kind (n, k) if #L(P ) = n and any edge of P divides L(P ) into two sets such that at least one of them has cardinality greater than or equal to k. In [2] we proved that, if 3 ≤ k ≤ n − 1, given a l-treelike family of real numbers, {D I } I∈( {1,...,n} k ) , there exists exactly one internal-nonzero-weighted pseudostar P of kind (n, k) with leaves 1, ..., n and no vertices of degree 2 such that D I (P) = D I for any I. Here we associate to any pseudostar of kind (n, k) with leaves 1, ..., n a hierarchy on {1, ..., n} with clusters of cardinality between 2 and n − k and, by using this association and by pushing forward the ideas in [9] and [10], we get a theorem (Theorem 18) characterizing l-treelike dissimilarity families; consequently, we obtain also a characterization of p-l-treelike dissimilarity families (see Remark 19).…”

Moduli Space of Families of Positive (n − 1)-Weights

“…. , n. (It is a mainly open the problem to characterize when a family of positive numbers is the family of the k-weights of a weighted tree, and 3 ≤ k ≤ n − 2, see for positive results [8,11,12].) The condition found by Baldisserri and Rubei hinges upon the following inequality (n − 2)D 1,...,î,...,n ≤ n j=1,j =i D 1,...j,...,n .…”

A characterization of dissimilarity families of trees

“…Also the study of general weighted trees can be interesting and, in [3], Bandelt and Steel proved a result, analogous to Theorem 3, for general weighted trees: An easy variant of the theorems above is the following: In fact, if the 4-point condition holds, in particular the relaxed 4-point condition holds, so by Theorem 4, there exists a weighted tree T with leaves 1, ..., n and with 2-weights equal to the D I ; it is easy to see that, since the 4-point condition holds, the weights of the internal edges of T are nonnegative; by contracting the edges of weight 0, we get an ip-weighted tree with leaves 1, ..., n and with 2-weights equal to the D I . For higher k the literature is more recent, see [1], [4], [9], [10], [11], [12], [14], [15], [16]. Three of the most important results for higher k are the following:…”