An additional adjacency between two levels in a complete binary tree minimizing total path length

This paper proposes a problem of adding relation to an organization structure which is a complete binary tree of height H. When one edge between a node with a depth M and its descendant with a depth N is added, an optimal pair of depth (M,N)* = (0,2) for H = 2,3,4,5 and (M,N)* = (0,4) for H = 6,7, ··· are shown. (M,N)* is obtained by maxmizing the total shortening path length which is the sum of shortening lengths of shortest paths between every pair of all nodes.