Automated Counting and Statistical Analysis of Labeled Trees with Degree Restrictions

TL;DR

This paper develops automated methods to enumerate and analyze labeled trees with specific degree restrictions, providing new proofs and detailed statistical properties including asymptotic normality and covariance structures.

Contribution

It introduces automated enumeration techniques for degree-restricted labeled trees and offers detailed statistical analysis including asymptotic distributions and covariance calculations.

Findings

Enumeration of degree-restricted labeled trees is fully automated.

Number of vertices with a given degree is asymptotically normal.

Pairs of degree counts are jointly asymptotically normal, with some independence.

Abstract

Arthur Cayley famously proved that there are n to the power n-2 labeled trees on n vertices. Here we go much further and show how to enumerate, fully automatically, labeled trees such that every vertex has a number of neighbors that belongs to a specified finite set, and also count trees where the number of neighbors is not allowed to be in a given finite set. We also give detailed statistical analysis, and show that in the sample space of labeled trees with n vertices, the random variable "number of vertices with d neighbors" is asymptotically normal, and for any different degrees, are jointly asymptotically normal, but of course, not independently so (except for the pair (1,3), i.e. the number of leaves and the number of degree-3 vertices, where there are asymptotically independent). We also give new proofs to Amram Meir and John Noon's expressions for the limiting expectation and variance for these, and derive an explicit expression for the covariance.