The minimum asymptotic density of binary caterpillars

TL;DR

This paper investigates the asymptotic density of binary caterpillars within large rooted d-ary trees, providing an exact formula for the limit inferior of this density as the trees grow infinitely large.

Contribution

It derives an exact formula for the minimum asymptotic density of binary caterpillars in large rooted d-ary trees, extending previous results on tree densities.

Findings

Limit inferior of binary caterpillar density is zero unless the tree is a binary caterpillar.

Provides an explicit formula for the limit inferior of the density in large trees.

Enhances understanding of tree structure distributions in combinatorial tree theory.

Abstract

Given $d\geq 2$ and two rooted $d$-ary trees $D$ and $T$ such that $D$ has $k$ leaves, the density $\gamma(D,T)$ of $D$ in $T$ is the proportion of all $k$-element subsets of leaves of $T$ that induce a tree isomorphic to $D$, after erasing all vertices of outdegree $1$. In a recent work, it was proved that the limit inferior of this density as the size of $T$ grows to infinity is always zero unless $D$ is the $k$-leaf binary caterpillar $F^2_k$ (the binary tree with the property that a path remains upon removal of all the $k$ leaves). Our main theorem in this paper is an exact formula (involving both $d$ and $k$) for the limit inferior of $\gamma(F^2_k,T)$ as the size of $T$ tends to infinity.