Even and odd plane labelled bipartite trees

TL;DR

This paper investigates the classification of plane labelled bipartite trees into even and odd categories based on a geometric partition, providing insights into their structural properties.

Contribution

It introduces a geometric partition of bipartite trees into even and odd subsets, revealing a new structural understanding of these trees.

Findings

Partition into even and odd trees based on vertex count parity

Clear geometric interpretation of the even-odd classification

Structural properties of bipartite trees related to their labelling

Abstract

Let $T(n,m)$ be the set of all plane labelled bipartite trees with $n$ white vertices and $m$ black. If the number $n+m$ of vertices is even, then the set $T(n,m)$ is a union of two disjoint subsets --- subset od "even" trees and subset of "odd" trees. This partition has a clear geometric meaning.