Linear-time Algorithms for Construction of Optimal Numerations of Oriented Trees

The solution of a wide range of tasks is connected with the placement of objects of different nature in elements of certain structures. The need for such actions arises, for example, when designing the location of the technological equipment of an enterprise, placing electronic equipment on boards, etc. As a rule, the objects to be placed have a certain set of relationships that can often be determined using a graph. In this case, the task is to find a numbering of the vertices of the graph that would bring the greatest value to a certain functional defined on the set of numberings. In the general case, for the problem of numbering graphs there is no effective solution algorithm. In this situation, it seems appropriate to study particular cases of numbering problems for limited classes of graphs (rooteed trees) for which there are algorithms that work in linear time.