On some relations between 2-trees and tree metrics

A tree function (TF) t on a ÿnite set X is a real function on the set of the pairs of elements of X satisfying the four-point condition: for all distinct x; y; z; w ∈ X; t(xy)+t(zw)6 max{t(xz)+ t(yw); t(xw) + t(yz)}. Equivalently, t is representable by the lengths of the paths between the leaves of a valued tree Tl. TFs are a straightforward generalization of the tree dissimilarities and tree metrics of the literature. A graph is a 2-tree if it belongs to the following class Q: an edge-graph belongs to Q: if ∈ Q and yz is an edge of , then the graph obtained by the addition to of a new vertex x adjacent to y and z belongs to Q. These graphs, and the more general k-trees, have been studied in the literature as generalizations of trees. It is ÿrst explicited here how to make a TF t ; d correspond to any positively valued 2-tree d on X . Then, given a tree dissimilarity t, the set Q(t) of the 2-trees such that t = t ; t is studied. Any element of Q(t) gives a way of summarizing t b...

• Type: Journal
• Year: 1998
• Where: DM
• Authors: Bruno Leclerc, Vladimir Makarenkov
• Keywords: DM 1998, Tree, Tree Dissimilarities, Tree Function