On several properties of plane-oriented recursive trees

TL;DR

This paper explores various properties of plane-oriented recursive trees, including degree distribution, Zagreb index, and asymptotic behavior, providing exact formulas, moments, and conjectures about their statistical laws.

Contribution

It offers new exact probability distributions, moments, and conjectures for properties of PORTs, advancing understanding of their structural and probabilistic characteristics.

Findings

Exact degree distribution for fixed nodes

First two moments of Zagreb index calculated

Evidence against Gaussian asymptotic law for Zagreb index

Abstract

In this paper, several properties of plain-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and variance of the degree variable via a \Polya\ urn approach. In addition, we look into a topological index, the Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index by using recurrence methods. We also provide several evidence in favor of our conjecture that the Zagreb index of PORTs do not follow a Gaussian law asymptotically. Lastly, we determine the limiting degree distribution in Poissonized PORTs, and show that it is exponential after being properly scaled.