Closed formula for counting binary tree-child networks

Derive a closed formula for the number of binary tree-child networks with n leaves and k reticulations in general.

Background

Tree-child networks are a major class of phylogenetic networks because they are both mathematically tractable and potentially reconstructible from leaf data. Although several methods have enumerated restricted cases and studied asymptotic behavior, the general enumeration problem remains unresolved.

The paper studies spinal tree-child networks as a particularly structured subclass that may serve as a base case for addressing the broader enumeration problem. Here, n denotes the number of leaves and k the number of reticulations.