Universal rooted phylogenetic tree shapes and 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 $n\le 11$. 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.

18 pages, 14 figures