Enumeration of one class of plane weighted trees

TL;DR

This paper introduces a method to enumerate a specific class of plane weighted trees characterized by positive integer weights and a bipartite structure, focusing on counting non-isotopic trees with given vertex weights.

Contribution

It provides a novel computational approach to determine the number of non-isotopic plane weighted trees with specified white and black vertex weights.

Findings

Developed a method for counting trees of a given type.

Applied the method to enumerate trees with specified weight distributions.

Enhanced understanding of the combinatorial structure of weighted trees.

Abstract

By weighted tree we understand such connected tree,that: a) each its vertex and each edge have a positive integer weight; b) the weight of each vertex is equal to the sum of weights of outgoing edges. Each tree has a binary structure --- we can color its vertices in two colors, black and white so, that adjacent vertices have different colors. A type is a set of pairwise non-isotopic plane weighted trees with a given list of weights of white vertices and a given list of weights of black vertices. In this work we present a method for computing the cardinality of a given type.