The sum of the distances between the leaves of a tree and the 'semi-regular' property

Various topological indices have been put forward in different studies, from biochemistry to pure mathematics. Among them, the Wiener index, the number of subtrees, and the Randi index have received great attention from mathematicians. In the study of extremal problems regarding these indices among trees, one interesting phenomenon is that they share the same extremal tree structures. Much effort was devoted to the study of the correlations between these various indices. In this note we provide a common characteristic (the 'semi-regular' property) of these extremal structures, with respect to the above mentioned indices, among trees with a given maximum degree. This observation leads to a more unified approach for characterizing these extremal structures. As an application/example, we illustrate the idea by studying the extremal trees, regarding the sum of distances between all pairs of leaves of a tree, a new index, which recently appeared in phylogenetic tree reconstruction, and the study of the neighborhood of trees.