arXiv subjects

Kristina Wicke

Publications and source records attributed to Kristina Wicke.

At least 19 recordsLinked to original sources

arXiv2026

Properties of biodiversity indices that incorporate future extinction risk

The loss of biodiversity due to the likely widespread extinction of species in the near future is a focus of current concern in conservation biology. One approach to measure the impact of this extinction is based on the predicted loss of phylogenetic diversity. These predictions have become a focus of the Zoological Society of London's `EDGE2' program for quantifying biodiversity loss and involves considering the HED (heightened evolutionary distinctiveness) and HEDGE (heightened evolutionary distinctiveness and globally endangered) indices which are based on phylogenetic diversity on a tree. Here, we show how to generalise the HED(GE) indices by expanding their application to more general settings (to phylogenetic networks, to feature diversity on discrete traits, and to arbitrary biodiversity measures). We provide a simple and explicit description of the mean and, importantly, the variance of such measures, and illustrate our results by an application to the phylogeny and a small set of features for all 27 extant Crocodilians.

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arXiv2026

Phylogeny-based metrics of biodiversity: concepts and methods

The concept of valuing evolutionary history has a long tradition and was formalized by several research groups in the early 1990s, each aiming to capture the products of evolution more comprehensively than species richness alone. Preserving the "Tree of Life" has since become a central, if sometimes contested, goal in conservation biology. Faith's phylogenetic diversity (PD), defined as the sum of the edge lengths of a rooted tree connecting a set of focal entities, is the most widely used metric in this context and underpins an extensive methodological framework. This framework includes widely used species-specific indices such as EDGE, as well as approaches that quantify expected species-specific contributions to PD such as EDGE2. This chapter focuses on the development of this framework of phylogeny-based conservation metrics. We review the development of species-specific indices, highlighting their strengths and limitations, and trace their progression from simple formulations to game-theoretic approaches and more abstract generalizations. We also discuss how these indices are applied in conservation practice, with particular emphasis on the EDGE framework. We further discuss the relationship between the edge lengths of a phylogeny and the evolutionary features they are intended to represent. Finally, we highlight recent advances in the field and identify areas for future research.

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arXiv2026

Enumerating monophyletic characters in mathematical phylogenetics

Grouping species according to their phylogenetic relationships often results in different groups than grouping them according to their shared traits. Monophyletic groups play an important role in this regard, as they are groups of species sharing the same trait and being uniquely defined by a joint phylogenetic subtree. This immediately leads to the question of how to identify possible monophyletic groups in characters, which assign each present-day species a certain trait and which are typically used for phylogenetic tree reconstruction. In our manuscript, we provide a general formula to quantify how many different characters are monophyletic on any given tree and provide simple formulae for binary characters and for certain tree shapes. We also investigate relations between monophyly and the well-known phylogenetic tree reconstruction criterion maximum parsimony by providing a linear-time algorithm which determines the parsimony score together with the monophyly type of a character on a tree.

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arXiv2026

On Agreement Subtrees in Multiple Phylogenetic Trees

Snir and Yuster [Discrete Appl. Math. 347 (2026) 160--171] asked for the least number $h(k)$ such that $k$ unrooted binary phylogenetic trees on the same $h(k)$ leaves always share a common quartet. We give a new upper bound for the $k$-tree version of the Maximum Agreement Subtree problem, namely an upper bound for the number of leaves, on which $k$ unrooted binary phylogenetic trees always share a common induced binary subtree on $n$ leaves, which is a four-times iterated exponential function. For $h(k)$, this implies a four-times iterated exponential upper bound. We also set an exponential lower bound for $h(k)$.

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arXiv2026

Revisiting a random model of lateral gene transfer in phylogenetics

Processes such as incomplete lineage sorting (ILS) and reticulate evolution (arising, for example, from hybridization and lateral gene transfer (LGT)) are known to cause discordance between gene trees and species trees, complicating phylogenetic inference. While the multispecies coalescent model has led to a rich theoretical understanding of ILS, probabilistic models for LGT have received comparatively less attention. Here, we revisit a basic LGT model in which random LGT events occur according to a Poisson process with a constant transfer rate. Focusing on the simplest cases of two and three species, we derive gene tree and site pattern probabilities and discuss their implications for model identifiability. We also address the question of whether LGT and ILS can be distinguished from one another under these probabilistic models. We discuss empirical applications and outline directions for future work.

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arXiv2026

A height-based metaconcept for rooted tree balance and its implications for the $B_1$ index

Tree balance has received considerable attention in recent years, both in phylogenetics and in other areas. Numerous (im)balance indices have been proposed to quantify the (im)balance of rooted trees. A recent comprehensive survey summarized this literature and showed that many existing indices are based on similar underlying principles. To unify these approaches, three general metaconcepts were introduced, providing a framework to classify, analyze, and extend imbalance indices. In this context, a metaconcept is a function $Φ_f$ that depends on another function $f$ capturing some aspect of tree shape. In this manuscript, we extend this line of research by introducing a new metaconcept based on the heights of the pending subtrees of all inner vertices. We provide a thorough analysis of this metaconcept and use it to answer open questions concerning the well-known $B_1$ balance index. In particular, we characterize the tree shapes that maximize the $B_1$ index in two cases: (i) arbitrary rooted trees and (ii) binary rooted trees. For both cases, we also determine the corresponding maximum values of the index. Finally, while the $B_1$ index is induced by a so-called third-order metaconcept, we explicitly introduce three new (im)balance indices derived from the first- and second-order height metaconcepts, respectively, thereby demonstrating that pending subtree heights give rise to a variety of novel (im)balance indices.

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arXiv2025

Distinguishing Phylogenetic Level-2 Networks with Quartets and Inter-Taxon Quartet Distances

The inference of phylogenetic networks, which model complex evolutionary processes including hybridization and gene flow, remains a central challenge in evolutionary biology. Until now, statistically consistent inference methods have been limited to phylogenetic level-1 networks, which allow no interdependence between reticulate events. In this work, we establish the theoretical foundations for a statistically consistent inference method for a much broader class: semi-directed level-2 networks that are outer-labeled planar and galled. We precisely characterize the features of these networks that are distinguishable from the topologies of their displayed quartet trees. Moreover, we prove that an inter-taxon distance derived from these quartets is circular decomposable, enabling future robust inference of these networks from quartet data, such as concordance factors obtained from gene tree distributions under the Network Multispecies Coalescent model. Our results also have novel identifiability implications across different data types and evolutionary models, applying to any setting in which displayed quartets can be distinguished.

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arXiv2025

A sharp lower bound for the number of phylogenetic trees displayed by a tree-child network

A normal (phylogenetic) network with $k$ reticulations displays $2^k$ phylogenetic trees. In this paper, we establish an analogous result for tree-child (phylogenetic) networks with no underlying $3$-cycles. In particular, we show that a tree-child network with $k\ge 2$ reticulations and no underlying $3$-cycles displays at least $2^{k/2}$ phylogenetic trees if $k$ is even and at least $\frac{3}{2\sqrt{2}}2^{k/2}$ if $k$ is odd. Moreover, we show that these bounds are sharp and characterise the tree-child networks that attain these bounds.

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arXiv2025

Reconstruction of caterpillar tanglegrams

A tanglegram consists of two rooted binary trees with the same number of leaves and a perfect matching between the leaves of the trees. Given a size-$n$ tanglegram, i.e., a tanglegram for two trees with $n$ leaves, a multiset of induced size-$(n-1)$ tanglegrams is obtained by deleting a pair of matched leaves in every possible way. Here, we analyze whether a size-$n$ tanglegram is uniquely encoded by this multiset of size-$(n-1)$ tanglegrams. We answer this question affirmatively in the case that at least one of the two trees of the tanglegram is a caterpillar tree.

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arXiv2025

Metaconcepts of rooted tree balance

Measures of tree balance play an important role in many different research areas such as mathematical phylogenetics or theoretical computer science. Typically, tree balance is quantified by a single number which is assigned to the tree by a balance or imbalance index, of which several exist in the literature. Most of these indices are based on structural aspects of tree shape, such as clade sizes or leaf depths. For instance, indices like the Sackin index, total cophenetic index, and $\widehat{s}$-shape statistic all quantify tree balance through clade sizes, albeit with different definitions and properties. In this paper, we formalize the idea that many tree (im)balance indices are functions of similar underlying tree shape characteristics by introducing metaconcepts of tree balance. A metaconcept is a function $Φ_f$ that depends on a function $f$ capturing some aspect of tree shape, such as balance values, clade sizes, or leaf depths. These metaconcepts encompass existing indices but also provide new means of measuring tree balance. The versatility and generality of metaconcepts allow for the systematic study of entire families of (im)balance indices, providing deeper insights that extend beyond index-by-index analysis.

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arXiv2025

Order-Dependent Dissimilarity Measures on Phylogenetic Trees

Ordered leaf attachment, Phylo2Vec, and HOP are three recently introduced vector representations for rooted phylogenetic trees where the representation is determined by an ordering of the underlying leaf set X. Comparing the vectors of two rooted phylogenetic X-trees T and T' for a fixed ordering on X leads to polynomial-time computable measure for the dissimilarity of T and T', albeit dependent on the choice of the leaf ordering. For each of ordered leaf attachment, Phylo2Vec, and HOP, we compare this measure with the rooted subtree prune and regraft distance (rSPR), the hybrid number, and the temporal tree-child hybrid number of T and T'. Although there is no direct relationship between rSPR and any of the three vector-based measures, we show that, when minimized over all orderings, the hybrid number is equivalent to HOP, and an upper bound on the other two. Moreover, when minimized over all orderings induced by common cherry-picking sequences of T and T', the temporal tree-child hybrid number of T and T' is equivalent to each of the three vector-based measures.

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arXiv2024

Tree balance in phylogenetic models

Tree shape statistics, particularly measures of tree (im)balance, play an important role in the analysis of the shape of phylogenetic trees. With applications ranging from testing evolutionary models to studying the impact of fertility inheritance and selection, or tumor development and language evolution, the assessment of measures of tree balance is important. Currently, a multitude of at least 30 (im)balance indices can be found in the literature, alongside numerous other tree shape statistics. This diversity prompts essential questions: How can we assist researchers in choosing only a small number of indices to mitigate the challenges of multiple testing? Is there a preeminent balance index tailored to specific tasks? This research expands previous studies on the examination of index power, encompassing almost all established indices and a broader array of alternative models, such as a variety of trait-based models. Our investigation reveals distinct groups of balance indices better suited for different tree models, suggesting that decisions on balance index selection can be enhanced with prior knowledge. Furthermore, we present the \textsf{R} software package \textsf{poweRbal} which allows the inclusion of new indices and models, thus facilitating future research on the power of tree shape statistics.

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arXiv2024

Bounding the softwired parsimony score of a phylogenetic network

In comparison to phylogenetic trees, phylogenetic networks are more suitable to represent complex evolutionary histories of species whose past includes reticulation such as hybridisation or lateral gene transfer. However, the reconstruction of phylogenetic networks remains challenging and computationally expensive due to their intricate structural properties. For example, the small parsimony problem that is solvable in polynomial time for phylogenetic trees, becomes NP-hard on phylogenetic networks under softwired and parental parsimony, even for a single binary character and structurally constrained networks. To calculate the parsimony score of a phylogenetic network $N$, these two parsimony notions consider different exponential-size sets of phylogenetic trees that can be extracted from $N$ and infer the minimum parsimony score over all trees in the set. In this paper, we ask: What is the maximum difference between the parsimony score of any phylogenetic tree that is contained in the set of considered trees and a phylogenetic tree whose parsimony score equates to the parsimony score of $N$? Given a gap-free sequence alignment of multi-state characters and a rooted binary level-$k$ phylogenetic network, we use the novel concept of an informative blob to show that this difference is bounded by $k+1$ times the softwired parsimony score of $N$. In particular, the difference is independent of the alignment length and the number of character states. We show that an analogous bound can be obtained for the softwired parsimony score of semi-directed networks, while under parental parsimony on the other hand, such a bound does not hold.

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arXiv2024

Phylogenetic diversity indices from an affine and projective viewpoint

Phylogenetic diversity indices are commonly used to rank the elements in a collection of species or populations for conservation purposes. The derivation of these indices is typically based on some quantitative description of the evolutionary history of the species in question, which is often given in terms of a phylogenetic tree. Both rooted and unrooted phylogenetic trees can be employed, and there are close connections between the indices that are derived in these two different ways. In this paper, we introduce more general phylogenetic diversity indices that can be derived from collections of subsets (clusters) and collections of bipartitions (splits) of the given set of species. Such indices could be useful, for example, in case there is some uncertainty in the topology of the tree being used to derive a phylogenetic diversity index. As well as characterizing some of the indices that we introduce in terms of their special properties, we provide a link between cluster-based and split-based phylogenetic diversity indices that uses a discrete analogue of the classical link between affine and projective geometry. This provides a unified framework for many of the various phylogenetic diversity indices used in the literature based on rooted and unrooted phylogenetic trees, generalizations and new proofs for previous results concerning tree-based indices, and a way to define some new phylogenetic diversity indices that naturally arise as affine or projective variants of each other.

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arXiv2024

The weighted total cophenetic index: A novel balance index for phylogenetic networks

Phylogenetic networks play an important role in evolutionary biology as, other than phylogenetic trees, they can be used to accommodate reticulate evolutionary events such as horizontal gene transfer and hybridization. Recent research has provided a lot of progress concerning the reconstruction of such networks from data as well as insight into their graph theoretical properties. However, methods and tools to quantify structural properties of networks or differences between them are still very limited. For example, for phylogenetic trees, it is common to use balance indices to draw conclusions concerning the underlying evolutionary model, and more than twenty such indices have been proposed and are used for different purposes. One of the most frequently used balance index for trees is the so-called total cophenetic index, which has several mathematically and biologically desirable properties. For networks, on the other hand, balance indices are to-date still scarce. In this contribution, we introduce the \textit{weighted} total cophenetic index as a generalization of the total cophenetic index for trees to make it applicable to general phylogenetic networks. As we shall see, this index can be determined efficiently and behaves in a mathematical sound way, i.e., it satisfies so-called locality and recursiveness conditions. In addition, we analyze its extremal properties and, in particular, we investigate its maxima and minima as well as the structure of networks that achieve these values within the space of so-called level-$1$ networks. We finally briefly compare this novel index to the two other network balance indices available so-far.

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arXiv2024

On the maximum value of the stairs2 index

Measures of tree balance play an important role in different research areas such as mathematical phylogenetics or theoretical computer science. The balance of a tree is usually quantified in a single number, called a balance or imbalance index, and several such indices exist in the literature. Here, we focus on the stairs2 balance index for rooted binary trees, which was first introduced in the context of viral phylogenetics but has not been fully analyzed from a mathematical viewpoint yet. While it is known that the caterpillar tree uniquely minimizes the stairs2 index for all leaf numbers and the fully balanced tree uniquely maximizes the stairs2 index for leaf numbers that are powers of two, understanding the maximum value and maximal trees for arbitrary leaf numbers is an open problem in the literature. In this note, we fill this gap by showing that for all leaf numbers, there is a unique rooted binary tree maximizing the stairs2 index. Additionally, we obtain recursive and closed expressions for the maximum value of the stairs2 index of a rooted binary tree with $n$ leaves.

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arXiv2023

Tree balance indices: a comprehensive survey

Tree balance plays an important role in phylogenetics and other research areas, which is why several indices to measure tree balance have been introduced over the years. Nevertheless, a formal definition of what a balance index actually is and what makes it a useful measure of balance (or, in other cases, imbalance), has so far not been introduced in the literature. While the established indices all summarize the (im)balance of a tree in a single number, they vary in their definitions and underlying principles. It is the aim of the present manuscript to introduce formal definitions of balance and imbalance indices that classify desirable properties of such indices and to analyze and categorize established indices accordingly. In this regard, we review 19 established (im)balance indices from the literature, summarize their general, statistical and combinatorial properties (where known), prove numerous additional results and indicate directions for future research by making explicit open questions and gaps in the literature. We also prove that a few tree shape statistics that have been used to measure tree balance in the literature do not fulfill our definition of an (im)balance index, which might indicate that their properties are not as useful for practical purposes. Moreover, we show that five additional tree shape statistics from other contexts actually are tree (im)balance indices according to our definition. The manuscript is accompanied by the website \url{treebalance.wordpress.com} containing fact sheets of the discussed indices. Moreover, we introduce the software package \verb|treebalance| implemented in $\mathsf{R}$ that can be used to calculate all indices discussed.

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arXiv2023

Decks of rooted binary trees

We consider extremal problems related to decks and multidecks of rooted binary trees (a.k.a. rooted phylogenetic tree shapes). Here, the deck (resp. multideck) of a tree $T$ refers to the set (resp. multiset) of leaf induced binary subtrees of $T$. On the one hand, we consider the reconstruction of trees from their (multi)decks. We give lower and upper bounds on the minimum (multi)deck size required to uniquely encode a rooted binary tree on $n$ leaves. On the other hand, we consider problems related to deck cardinalities. In particular, we characterize trees with minimum-size as well as maximum-size decks. Finally, we present some exhaustive computations for $k$-universal trees, i.e., rooted binary trees that contain all $k$-leaf rooted binary trees as induced subtrees.

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