On the relation between weighted trees and tropical Grassmannians

In this article, we will prove that the set of 4-dissimilarity vectors of n-trees is contained in the tropical Grassmannian G4,n. We will also propose three equivalent conjectures related to the set of m-dissimilarity vectors of n-trees for the case m≥5. Using a computer algebra system, we can prove these conjectures for m=5.