A model of adding two edges in levels of a complete binary tree

This study proposes a model of adding two edges between nodes of the same level of a complete binary tree. Firstly we add one edge between the optimal two nodes with the optimal depth N* maximizing a total shortening distance. Secondly we add another edge between nodes with the same depth M (M = 1, 2, ..., H). The total shortening distance to obtain the optimal two nodes with the optimal depth M* maximizing a total shortening distance is formulated.


Introduction
This paper proposes a new model of adding relations in an organization structure [1,2].Relations between members in an organization are added to become effective in communication of information.
This study considers a pyramid organization structure which is a complete binary tree of height H (H = 2, 3, ...).A complete binary tree is a rooted tree in which all leaves have the same depth and all internal nodes have two children [3].Nodes and edges in the complete binary tree correspond to units and relations between units in the organization.Moreover the path between a pair of nodes in the complete binary tree is equivalent to the route of communication of information between a pair of units in the organization, and adding edges is equivalent to forming additional relations other than that between each superior and his direct subordinates [4][5][6][7].
We have proposed a model [4][5][6][7] of adding one edge between two nodes with the same depth N (N = 1, 2, ..., H) of a complete binary tree.In this model we have obtained the optimal two nodes with the optimal depth N * maximizing a total shortening distance.The total shortening distance is the sum of shortened lengths of shortest paths between every pair of all nodes by adding edge between two nodes [4][5][6][7].This means that the communication of information between every member in the organization becomes the most efficient.
This paper proposes a model of adding two edges between nodes of the same level of a complete binary tree.In this model firstly we add one edge between the optimal two nodes with the optimal depth N * maximizing a total shortening distance [4][5][6][7].Secondly we add another edge between nodes with the same depth M (M = 1, 2, ..., H).We formulate the total shortening distance to obtain the optimal two nodes with the optimal depth M * maximizing a total shortening distance.

A model of adding one edge [4-7]
This section formulates the total shortening distance when an edge between nodes with the same depth N (N = 1, 2, ..., H) in a complete binary tree of height H (H = 2, 3, ...) and obtains the optimal two nodes with the optimal depth N * maximizing a total shortening distance [4][5][6][7].
A new edge can be added between two nodes with the same depth N in a complete binary tree in N ways that lead to non-isomorphic graphs.Let R H (N, D) denote the total shortening distance by adding the new edge, where D (D = 0, 1, 2, ..., N -1) is the depth of the deepest common ancestor of the two nodes on which the new edge is incident.When the total shortening distance for the case of We formulate S H (N) in the following.
Let v 0 X and v 0 Y denote the two nodes on which the adding edge is incident and assume that , where k = 1, 2, ..., N -1.The total shortening distance can be formulated by adding up the following three sums of shortening distances: (i) the sum of shortening distances between every pair of nodes in V 0 X and nodes in V 0 Y , (ii) the sum of shortening distances between every pair of nodes in V 0 X and nodes in V k Y (k = 1, 2, ..., N -1) and between every pair of nodes in V 0 Y and nodes in V k X (k = 1, 2, ..., N -1) and (iii) the sum of shortening distances between every pair of nodes in V k X (k = 1, 2, ..., N -1) and nodes in V k Y (k = 1, 2, ..., N -1).
The sum of shortening distances between every pair of nodes in V 0 X and nodes in where W (h) denotes the number of nodes of a complete binary tree of height h (h = 0, 1, 2, ...).The sum of shortening distances between every pair of nodes in V 0 X and nodes in V k Y (k = 1, 2, ..., N -1) and between every pair of nodes in and the sum of shortening distances between every pair of nodes in V k X (k = 1, 2, ..., N -1) and nodes in where we define 0 0 1 x ¦ i and 0 From Equations ( 2), ( 3) and ( 4), the total shortening distance S H (N) for the case of D = 0 is formulated by .
Theorem 1 shows that the most efficient way of forming relations to two members in each level is that to two members which doesn't have common superiors except the top.

A model of adding two edges
This section proposes a new model of adding two edges between nodes of the same level of a complete binary tree.Firstly we add one edge between the optimal two nodes with the optimal depth N * in Section 2 of this paper.Secondly we add another edge between nodes with the same depth M (M = 1, 2, ..., H).The total shortening distance to obtain the optimal two nodes with the optimal depth M * maximizing a total shortening distance is formulated in this section.
Let n X and n Y denote the two nodes on which the optimal first adding edge with the optimal depth N * is incident.Let m X and m Y denote the two nodes on which the second adding edge with the depth M is incident.Let DDCA(u, v) denote the depth of the deepest common ancestor of node u and node v.
The total shortening distance is formulated in the following cases.