Sets of double and triple weights of trees

2007/12/18 by Rubei, Elena
#05C05 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.0712.3026

Abstract

Let T be a weighted tree with n leaves. Let D_i,j be the distance between the leaves i and j. Let D_i,j,k= (D_i,j + D_j,k +D_i,k)/2. We will call such numbers "triple weights" of the tree. In this paper, we give a characterization, different from the previous ones, for sets indexed by 2-subsets of a n-set to be double weights of a tree. By using the same ideas,we find also necessary and sufficient conditions for a set of real numbers indexed by 3-subsets of an n-set to be the set of the triple weights of a tree with n leaves. Besides we propose a slight modification of Saitou-Nei's Neighbour-Joining algorithm to reconstruct trees from the data D_i,j.

Related

• On dissimilarity vectors of (not necessarily positive) weighted trees
• Reconstructing fully-resolved trees from triplet cover distances
• Treelike families of multiweights
• Combinatorial properties of triplet covers for binary trees
• Minimum triplet covers of binary phylogenetic X-trees