Conjectured recursion for binary tree-child network counts

Prove that the number $|TC_{n,k}|$ of binary tree-child networks with k reticulations and n leaves satisfies the recursion $(n-k)|TC_{n,k}|=(n+1-k)(n-k)|TC_{n,k-1}|+n(2n+k-3)|TC_{n-1,k}|$.

Background

The paper identifies a conjectured recurrence as a harder open problem in the enumeration of tree-child networks. The recurrence concerns binary, non-degenerate tree-child networks with n leaves and k reticulations.

The authors suggest that enumerating spinal tree-child networks may provide a base case from which a recursion for the number of general tree-child networks could eventually be identified.

References

For instance, one conjecture is that the number $|TC_{n,k}|$ of binary tree-child networks with $k$ reticulations and $n$ leaves is given by the recursion:

— Counting spinal phylogenetic networks  (2502.14223 - Francis et al., 20 Feb 2025) in Section 1, Introduction; Equation (1)