Day 3

　　If the true evolutionary history of a group X of taxa has as its underlying graph a network N, then a summary of N can be given by its tree of blobs T. After undirecting and unrooting N, this tree T is obtained by contracting all 2-edge connected components to nodes, yielding the tree of blobs with high-degree nodes indicating the presence of reticulate evolutionary history. 

　　We introduce novel consistent statistical methods for inferring the tree of blobs from quartet concordance factors generated under the Network Multispecies Coalescent Model on N. These methods proceed by starting with a fully resolved tree onX and then contracting edges to obtain T, where an edge e is contracted if there is statistical evidence against e’s presence in T. 

　　In this talk we explain the ideas underlying our new method(s), describe some of the theoretical advances made by these developments, and give examples of applying these methods to simulated and empirical gene tree data. This is joint work with Cecile Ane, Hector Banos, and John Rhodes.

　　Cluster reduction is a powerful divide-and-conquer technique for the hybridization number problem, which is the problem taking a set of phylogenetic trees as input and asking for a phylogenetic network that displays the given trees and has a minimum reticulation number. Baroni, Semple and Steel showed in 2006 that cluster reduction is safe for two binary trees. This question was left open for more than two binary trees. In this talk, I will show that cluster reduction is not safe for multiple binary trees. I will also discuss various related questions. This is joint work with Mark Jones, Kaari Landry, Simone Linz, Mathias Weller and Norbert Zeh.

　　B cell lineage trees capture clonal relationships and evolutionary trajectories, yet their structural variability makes comparison and synthesis challenging. We introduce a combinatorial metric that integrates branch lengths and node abundances, with “ghost nodes” added to align trees defined on disjoint node sets. This metric supports clustering of lineage trees and the construction of representative supertrees, summarizing dominant phylogenetic signals. Applied to both simulated and experimental human B cell repertoires, our approach highlights key evolutionary patterns such as localized hypermutation hotspots and clonal expansions, allowing a systematic analysis of B cell evolution.

In recent years, significant progress has been made in both asymptotic and exact counting of phylogenetic networks. In this talk, I will present several results that simplify existing counting for two important classes of networks: tree-child networks and galled networks. A phylogenetic network is tree-child if no two reticulation nodes are siblings or in the parentchild relation, and it is galled if no reticulation node has parents in different tree components.

　　Time-consistent galled trees provide a simple class of rooted binary network structures that can be used to represent a variety of different biological phenomena. We study the asymptotic number of unlabeled time-consistent galled trees with n leaves (n tending to infinity) and a fixed number of galls and the analogous problem for labeled time-consistent galled trees. Our approach to these problems is by means of generating functions and singularity analysis. This is joint work with Lily Agranat-Tamir, Michael Fuchs and Noah A. Rosenberg.

　　Understanding the size of phylogenetic network classes and the typical shape of a random network from a fixed class has been one of the major research focuses in phylogenetics over the last couple of years. In this talk, we introduce two subclasses of the (recently introduced) class of semi-simplex phylogenetic networks, namely, semi-simplex tree-child networks and semi-simplex galled trees. We consider their sizes relative to the (known) sizes of general tree-child networks and galled trees, respectively, and the limit laws for parameters of random networks from these classes.

　　Spinal tree-child networks form a rigid subclass of phylogenetic networks with a rich combinatorial structure (see Francis and Hendriksen). We introduce a new encoding for spinal networks via a restricted class of words related to those of Pons and Batle, yielding a direct bijection and an explicit enumeration. This approach recovers known counting formulas by linking the words to set partitions, and provides an explicit translation between different encodings. Finally, we present a combinatorial specification leading to closed formulas and generating functions, and indicate extensions to spinal families. This is joint work with Pau Vives, Anna de Mier, and Gabriel Cardona.

Given a phylogenetic tree, one can iteratively reduce it using so-called cherrypicking sequences. For a subtree T′ of a tree T, one can show that T′ is displayed by T if and only if there exists a cherry-picking sequence of T which reduces T′. This property does not fully extend to the classes of tree-child or orchard networks, making it natural to consider the following problem for trees:

Covering Number: Given a tree T, what is the minimum number of cherry-picking sequences of T needed to reduce every subtree of T?

　　If T is binary, then the problem can be solved recursively. If T is non-binary, in particular if T is a directed star, then it can be shown that the problem is equivalent to another combinatorial optimization problem, whose computational complexity remains open.

　　Finally, we show that the covering number of a tree provides a lower bound for the orchard hybridization sequence problem on certain input trees, linking our results to the problem of finding minimum-length orchard sequences that reduce all input trees.

　　This is joint work with Bálint Kollman and Takatora Suzuki.