Some extremal ratios of the distance and subtree problems in binary trees

Among many topological indices of trees the sum of distances σ(T) and the number of subtrees F(T) have been a long standing pair of graph invariants that are well known for their negative correlation. That is, among various given classes of trees, the extremal structures maximizing one usually minimize the other, and vice versa. By introducing the “local” versions of these invariants, σ_T(v) for the sum of distance from v to all other vertices and F_T(v) for the number of subtrees containing v, extremal problems can be raised and studied for vertices within a tree. This leads to the concept of “middle parts” of a tree with respect to different indices. A challenging problem is to find extremal values of the ratios between graph indices and corresponding local functions at middle parts or leaves. This problem also provides new opportunities to further verify the the correlation between different indices such as σ(T) and F(T). Such extremal ratios, along with the extremal structures, were studied and compared for the distance and subtree problems for general trees (Barefoot, 1997; Székely and Wang, 2013, 2014). In this paper, this study is extended to binary trees, a class of trees with numerous practical applications in which the extremal ratio problems appear to be even more complicated. After justifying some basic properties on the distance and subtree problems in trees and binary trees, characterizations are provided for the extremal structures achieving two extremal ratios in binary trees of given order. The generalization of this work to k-ary trees is also briefly discussed. The findings are compared with the previous established extremal structures in general trees. Lastly some potential future work is mentioned.