The Problem of the Existence of a Tree with a Characteristic Vector of Node Vertices

Abstract

The paper presents the problem of the existence of a tree with certain numerical characteristics. It is clear that if a tree is given, it is possible to determine the number of node vertices of the tree and leaves, as well as to determine their degrees. Thus, for a tree, you can define a set of pairs whose coordinates are numbers corresponding to the number of node vertices and their degrees. We can form the inverse problem: we give pairs of natural numbers whose second coordinates are greater than 1, and we should determine whether there is at least one tree that the numbers of its node vertices and their degrees coincide with these pairs. The solution to this problem is presented in this paper.