Enumeration of one class of plane weighted trees

We present an enumeration formula for weighted trees, i.e., trees where vertices and edges have weights (a weight is a positive integer) and the weight of each vertex is equal to the sum of the weights of the edges incident to it. Each tree has a binary structure: we can color its vertices in two colors, black and white, so that adjacent vertices have different colors. In this work, the following problem is considered: enumerate weighted plane trees with given sets of weights of black and white vertices.