Self-transformation of trees with a bounded degree of vertices to minimize or maximize the Wiener index

We consider a distributed network whose communication graph is a non-oriented tree. It is assumed that the network itself can change its topology. For this, an extremely local atomic transformation is proposed — the addition of an edge connecting the different ends of two adjacent edges, and the simultaneous removal of one of these edges. This transformation is performed by "command" from the vertex of the tree, namely, the common vertex of two adjacent edges. It is shown that any other tree can be obtained from any tree using only atomic transformations. If trees with degree bounded $d$ ($d>2$) are considered, then the transformation does not violate this restriction. As an example of the goal of such a transformation, the tasks of maximizing and minimizing the Wiener index of a tree with a bounded degree of vertices without changing the set of its vertices are considered. Corresponding distributed algorithms are proposed and linear estimates of their complexity are proved.