Universal rooted phylogenetic tree shapes and universal tanglegrams

TL;DR

This paper establishes bounds on the minimal size of universal rooted binary trees and tanglegrams that contain all possible configurations for a given number of leaves or size, advancing understanding of their combinatorial properties.

Contribution

It provides the first explicit bounds for the minimal size of universal rooted binary trees and tanglegrams, including exact computations for small cases and generalizations to d-ary structures.

Findings

Omega(n log n) lower bound for universal rooted binary trees

O(n^2) upper bound for universal rooted binary trees

Omega(n^2) lower bound for universal tanglegrams

Abstract

We provide an $\Omega(n\log n) $ lower bound and an $O(n^2)$ upper bound for the smallest size of rooted binary trees (a.k.a. phylogenetic tree shapes), which are universal for rooted binary trees with $n$ leaves, i.e., contain all of them as induced binary subtrees. We explicitly compute the smallest universal trees for $n\leq 11$. We also provide an $\Omega(n^2) $ lower bound and an $O(n^4)$ upper bound for the smallest size of tanglegrams, which are universal for size $n$ tanglegrams, i.e., which contain all of them as induced subtanglegrams. Some of our results generalize to rooted $d$-ary trees and to $d$-ary tanglegrams.