Protected Vertices in Motzkin trees

TL;DR

This paper derives recurrence relations to compute the asymptotic probabilities of vertices being protected or balanced in Motzkin trees, providing bounds for these probabilities and expected ranks.

Contribution

It introduces recurrence relations for analyzing protected and balanced vertices in Motzkin trees, offering new probabilistic bounds and expectations.

Findings

Recurrence relations for protected vertices' probabilities

Bounds for balanced vertices' probabilities and ranks

Quantitative analysis of vertex properties in Motzkin trees

Abstract

In this paper we find recurrence relations for the asymptotic probability a vertex is $k$ protected in all Motzkin trees. We use a similar technique to calculate the probabilities for balanced vertices of rank $k$. From this we calculate upper and lower bounds for the probability a vertex is balanced and upper and lower bounds for the expected rank of balanced vertices.