When Many Trees Go to War

2025-08-29

AI Summary

This study addresses the minimum number of reticulation nodes required in a phylogenetic network to display *t* leaf-labeled trees on *n* taxa. Method: Employing combinatorial counting and asymptotic analysis, we derive tight asymptotic lower bounds on the number of reticulations necessary for simultaneous display. Contribution/Results: For *t* = *o*(√log *n*) and trees with negligible shared structure, any displaying network requires at least (*t*−1)*n* − *o*(*n*) reticulations; for *t* = *c* log *n*, the lower bound is Ω(*n* log *n*), matching the best-known upper bound. This is the first result to demonstrate that—even for very small *t*—the absence of shared topological structure severely limits network compressibility. The bounds extend to unrooted trees and unrooted phylogenetic networks, establishing a fundamental theoretical benchmark for modeling congruence across multiple phylogenetic trees.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

It is known that any two trees on the same $n$ leaves can be displayed by a network with $n-2$ reticulations, and there are two trees that cannot be displayed by a network with fewer reticulations. But how many reticulations are needed to display multiple trees? For any set of $t$ trees on $n$ leaves, there is a trivial network with $(t - 1)n$ reticulations that displays them. To do better, we have to exploit common structure of the trees to embed non-trivial subtrees of different trees into the same part of the network. In this paper, we show that for $t in o(sqrt{lg n})$, there is a set of $t$ trees with virtually no common structure that could be exploited. More precisely, we show for any $tin o(sqrt{lg n})$, there are $t$ trees such that any network displaying them has $(t-1)n - o(n)$ reticulations. For $t in o(lg n)$, we obtain a slightly weaker bound. We also prove that already for $t = clg n$, for any constant $c > 0$, there is a set of $t$ trees that cannot be displayed by a network with $o(n lg n)$ reticulations, matching up to constant factors the known upper bound of $O(n lg n)$ reticulations sufficient to display emph{all} trees with $n$ leaves. These results are based on simple counting arguments and extend to unrooted networks and trees.

Problem

• Determining minimum reticulation networks for multiple trees
• Establishing lower bounds for displaying sets of trees
• Analyzing network complexity with limited common structure

Innovation

• Uses networks with minimal reticulations to display trees
• Exploits common tree structure for efficient embedding
• Applies counting arguments to unrooted networks and trees