Four Point Condition

The four point condition is a central concept in describing distances between leaves of a tree. If a tree diagram is drawn connecting a set of points, then for each subset of four leaves of the tree \(x\), \(y\), \( z\), and \(t\), the distances between the points, measured as path lengths on the tree, must satisfy \(d(x,y)+d(z,t) \leq \max\{d(x,z)+d(y,t), d(x,t)+d(y,z)\}\). Moreover, if distances between points are specified such that all subsets of four points in set satisfy the four point condition, then the points can be represented as leaves on a tree. In a four-pointed tulip tree leaf, the four “points” of a leaf can represent four vertices in the four point condition. The distances among the four points satisfy the condition, as the points are connected through the tree-like vein structure of the leaf.