A tropical interpretation of m-dissimilarity maps

Abstract

Let T be a weighted tree with n numbered leaves and let D = (D (i, j))i, j be its distance matrix, so D (i, j) is the distance between the leaves i and j. If m is an integer satisfying 2 ≤ m ≤ n, we prove a tropical formula to compute the m-dissimilarity map of T (i.e. the weights of the subtrees of T with m leaves), given D. For m = 3, we present a tropical description of the set of m-dissimilarity maps of trees. For m = 4, a partial result is given

The split decomposition of a k-dissimilarity map

“…The tropical Grassmannian (see [22]) is a subset of the Dressian. It was shown by Iriarte [15] with methods developed by Bocci and Cools [2], and Cools [3] that for a weighted tree T , the weight function w D k T is a point in the tropical Grassmannian and hence in the Dressian. Corollary 6.3 now implies that w D k T is indeed in the interior of the cone of the Dressian spanned by the split weights w k S e for all splits S e corresponding to edges e of T .…”

Sets of Double and Triple Weights of Trees

“…In [1], Bocci and Cools gave a description of k-dissimiliarity maps of a tree and generalized Buneman's result for sets of real numbers D i, j, l indexed by 3-subsets of {1,..., n} coming from sets D i, j .…”

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:…”