Day 1

　　A number of encodings are in common use for phylogenetic trees and networks. These include those of mathematical simplicity, such as the encoding as a set of vertices and a set of edges, as well as some such as Newick format, that captures important information such as clade structures. In this talk I will talk about the general idea of encodings, and describe some recent results using encodings such as partitions, covers, folios, or graph substructures, and discuss how their different underlying paradigms might connect to the nature of their specific advantages. 

　　Tree-child networks play a central role in modeling reticulate evolutionary processes, yet their enumeration has long resisted effective methods. In 2021, Pons and Batle proposed a striking conjecture relating these networks to a simple class of constrained words, opening a new combinatorial perspective on the problem.

　　This talk is the first in a three-part series devoted to the history and resolution of the Pons–Batle conjecture. I will trace the development of ideas leading toward its solution: from Heinz’s OEIS sequence, through its reinterpretation in terms of tree-child networks, to the formulation of the conjecture and subsequent partial results. I will also introduce two generalizations of the underlying word model and explain how they naturally lead to Young tableaux with walls, providing a unifying combinatorial framework.
　　This lecture sets the stage for the next two talks. In the second talk, Michael will present an unpublished approach developed about a year and a half ago that yields partial progress toward the conjecture. In the third talk, Zhicong will describe a recently published method that completes the solution by uncovering an unexpected connection between the two  models via a summation identity.

　　The Pons-Batle Conjecture asserts that the family of specific words of length n and the family of tree-child networks with n nodes share the same cardinality. While a recent fully computational proof has established the validity of this statement, a combinatorial interpretation remains elusive. In this work, we approach the conjecture through a structural lens, demonstrating an equivalence between this problem and a correspondence between two families of lattice paths. By encoding this framework into generating functions and utilizing a recursive algorithm based on the number of reticulation nodes (indegree 2, outdegree 1), we are able to provide an alternative proof of the conjecture for all families with at most 100 reticulation nodes. Although a direct bijection remains an open problem, this lattice path perspective offers a meaningful step toward a combinatorial understanding of these structures.

　　Banderier, Marchal, and Wallner considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. In this talk, we prove a conjecture of Fuchs and Yu concerning the enumeration of two classes of three-row Young tableaux with walls. Combining with the work by Chang, Fuchs, Liu,Wallner, and Yu leads to the verification of a conjecture on tree-child networks proposed by Pons and Batle. This conjecture was regarded as a specific and challenging problem in the Phylogenetics community until it was finally resolved by the present work. This talk is based on joint work with Feihu Liu, Jiahang Liu, Jing Liu and Guoce Xin.

　　Phylogenetic trees and networks are directed acyclic graphs that represent evolutionary relationships between species in biology and related fields. Numerous classes of phylogenetic networks have been defined and investigated. If a network in some class C is restricted to a subset of its leaves, then the resulting induced network may lie outside C; by contrast, certain other classes of networks are ‘closed’ when we restrict to subsets of leaves (e.g. the class of phylogenetic trees). It turns out that any closed subclass of the class of phylogenetic trees is either all trees or a vanishingly small proportion of them (as the number of leaves grows). In this talk, we use asymptotic enumeration techniques to explore whether this dichotomy phenomenon holds for more general classes of phylogenetic networks.

　　Rooted tree-child networks are a popular class of phylogenetic networks that have favourable properties from a mathematical and algorithmic perspective without being overly simplistic. In this talk, we explore two classes of unrooted phylogenetic networks with the goal of expanding the notion of tree-child networks to the unrooted setting. First, we consider unrooted phylogenetic networks that are tree-child orientable and establish that they are computationally hard to recognise even in the binary case. Second, we introduce unrooted q-cuttable networks and show that they have several desirable properties which are similar to those of rooted tree-child networks. Joint work with Leo van Iersel, Mark Jones, and Norbert Zeh.

　　Phylogenetic networks, a generalization of phylogenetic trees, are rooted directed acyclic graphs representing reticulate evolutionary history. One natural approach to analyzing such networks is to recognize trees inside a network. Since the paper “Which Phylogenetic Networks are Merely Trees with Additional Arcs?” by Francis and Steel (2015), tree-based networks and their support trees (spanning trees with the same root and leaf set as a given network) have been extensively studied. Subsequently, Hayamizu (2021) established a structure theorem enabling optimal algorithms for various computational problems on support trees of rooted almost-binary (including binary) networks. However, not all networks are tree-based, and it is meaningful to consider spanning subgraphs beyond trees. In this talk, I present a generalization of the structure theorem to extend the theoretical framework for support trees to support networks. A key result is a direct-product characterization of each of three sets: all, minimal, and minimum support networks, for a given network. These characterizations yield closedform formulas for their cardinalities, revealing unexpected connections to Fibonacci, Lucas, Padovan, and Perrin numbers, and further lead to optimal algorithms for counting and listing the support networks of each type. Applications include a linear-time algorithm for finding a support network with the fewest reticulations. I also describe exact and heuristic algorithms for finding a support network with the minimum level, both running in exponential time but practical across a reasonably wide range of reticulation numbers. This is joint work with Takatora Suzuki and our conference paper is available at https://doi.org/10.4230/LIPIcs.WABI.2025.19

　　Quartet concordance factors (qCFs) are one of the key summaries of genomic data for phylogenetic inference under the multispecies coalescent. However, formulas for computing expected qCFs quickly become unwieldy for complex phylogenetic networks. Symbolic formulas are essential for proving parameter identifiability, understanding model behavior, and developing statistically sound inference methods beyond numerical simulation. In this talk, I introduce SymbolicQuartetCF.jl (symCF), a computational method written in the Julia programming language that automatically derives symbolic representations of expected qCFs for arbitrary phylogenies (i.e., both trees and networks). The package also supports direct export to computer algebra systems (e.g., Macaulay2), enabling downstream theoretical and mathematical analyses.