Counting rankings of tree-child networks

Rooted phylogenetic networks allow biologists to represent evolutionary relationships between present-day species by revealing ancestral speciation and hybridization events. A convenient and well-studied class of such networks are tree-child networks' and a ranking’ of such a network is a temporal ordering of the ancestral speciation and hybridization events. In this short note, we investigate the question of counting such rankings on any given binary (or semi-binary) tree-child network. We also consider a class of binary tree-child networks that have exactly one ranking, and investigate further the relationship between ranked-tree child networks and the class of `normal’ networks. Finally, we provide an explicit asymptotic expression for the expected number of rankings of a tree-child network chosen uniformly at random.

💡 Research Summary

The paper addresses the fundamental problem of enumerating temporal rankings of tree‑child phylogenetic networks, a class of rooted directed acyclic graphs that model speciation and hybridisation events. After reviewing basic graph‑theoretic notions (DAGs, topological orderings, δ(D) as the number of such orderings) the authors introduce the concept of “events”, equivalence classes of internal vertices defined by the relation R that groups vertices linked only by reticulation arcs. An event is either a branching event (single vertex) or a reticulation event (at least three vertices). A ranking is a map assigning integers 0,…,e_N‑1 to events, respecting two biological constraints: (T1) reticulation arcs connect vertices of equal rank, and (T2) tree arcs strictly increase rank.

The first major theoretical contribution is a characterization of rankable tree‑child networks. While it is known that every binary rankable tree‑child network is normal, the authors show that non‑binary examples can violate normality. Nevertheless, they prove that any separated tree‑child network that admits a ranking must be normal. The proof hinges on Lemma 2 (any directed path of length ≥ 3 in a temporally labelled network has non‑decreasing ranks) and Proposition 1, which rules out the existence of a “shortcut” arc (v₁, v_k) in a normal network because it would force contradictory rank equalities.

The core methodological advance is a transformation that converts the ranking problem into a classic counting‑topological‑order problem. For a given network N, the set of events d_N becomes the vertex set of a new DAG Ψ(N). An arc (ĥu, ĥv) is added whenever a tree arc in N connects some vertex of event ĥu to a vertex of event ĥv. The authors prove the bijection ψ(N)=δ(Ψ(N)), i.e., the number of rankings of N equals the number of topological orderings of Ψ(N). This equivalence is illustrated with several figures: in some cases Ψ(N) collapses to a rooted tree, allowing the use of the well‑known formula δ(T)=|V|!∏_v λ(v), where λ(v) counts descendants. Although counting topological orderings is #P‑hard for arbitrary DAGs, the special structure of Ψ(N) (stemming from the tree‑child and separation constraints) makes the computation tractable via dynamic programming in polynomial time.

Finally, the paper derives an asymptotic expression for the expected number of rankings of a uniformly random binary tree‑child network with n leaves and k reticulation vertices, denoted N_{n,k}. Using combinatorial enumeration of network shapes and the bijection above, they show
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