Trees with minimum number of infima closed sets

TL;DR

This paper characterizes rooted trees with the fewest infima closed sets for a given size, showing they are nearly complete binary trees, and provides an asymptotic estimate for their count.

Contribution

It identifies the structure of trees minimizing infima closed sets and offers an asymptotic estimate, answering a question posed by Klazar.

Findings

Trees with minimum infima closed sets are nearly complete binary trees.

Asymptotic estimate for the minimum number of infima closed sets in large trees.

The structure of optimal trees is characterized explicitly.

Abstract

Let $T$ be a rooted tree, and $V(T)$ its set of vertices. A subset $X$ of $V(T)$ is called an infima closed set of $T$ if for any two vertices $u,v\in X$, the first common ancestor of $u$ and $v$ is also in $X$. This paper determines the trees with minimum number of infima closed sets among all rooted trees of given order, thereby answering a question of Klazar. It is shown that these trees are essentially complete binary trees, with the exception of vertices at the last levels. Moreover, an asymptotic estimate for the minimum number of infima closed sets in a tree with $n$ vertices is also provided.