On the intercluster distance of a tree metric

For two vertex clusters of a tree metric, we show that the sum of the average intracluster distances is always less than or equal to twice of the average intercluster distance. We show the feature in a more general form of weighted distance. This feature provides a 2-approximation algorithm for the minimum average intercluster distance spanning tree problem, which is a generalization of the minimum routing cost spanning tree or minimum average distance spanning tree problem. The results in this paper can be further generalized to more than two clusters.