# The Proportion of Trees that are Linear

## TL;DR

This paper investigates enumeration problems related to linear trees, providing generating functions, asymptotic growth rates, and distributional properties, including a central limit theorem for the number of k-linear trees.

## Contribution

It introduces new generating functions, characterizes the asymptotic growth, and proves a central limit theorem for the distribution of k-linear trees.

## Key findings

- Derived generating functions for linear trees
- Established asymptotic growth rates of nonisomorphic linear trees
- Proved a central limit theorem for the distribution of k-linear trees

## Abstract

We study several enumeration problems connected to linear trees, a broad class which includes stars, paths, generalized stars, and caterpillars. We provide generating functions for counting the number of linear trees on $n$ vertices, characterize the asymptotic growth rate of the number of nonisomorphic linear trees, and show that the distribution of $k$-linear trees on $n$ vertices follows a central limit theorem.