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arXiv · 1804.05731

The minimum asymptotic density of binary caterpillars

Audace A. V. Dossou-Olory
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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.

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Audace A. V. Dossou-Olory. 2018-04-16. The minimum asymptotic density of binary caterpillars. https://arxiv.org/abs/1804.05731

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