Tree-child networks are an important class of phylogenetic network used to model reticulate evolutionary processes. These networks have attracted increasing attention from researchers with interests in both combinatorics and algorithms. A fundamental open problem posed by Pons and Batle asks whether the number TC_{n,k} of bicombining tree-child networks with n leaves and k reticulation nodes equals the number of certain constrained words, now called Pons-Batle words. In this paper, we confirm the conjecture for tree-child networks with a bounded number of reticulation nodes. Our approach is combinatorial and analytic. We introduce families of Young tableaux with walls and holes and construct explicit bijections with Pons-Batle words, yielding a direct combinatorial explanation of the identities. These tableaux encode structural features of the underlying networks, including the placement of reticulation nodes. By projecting them to decorated Dyck paths, we obtain algebraic generating functions with differential operators encoding step weights, leading to explicit recurrence relations and closed-form formulas for TC_{n,k}. Beyond finite verification for moderate k, the framework reveals an underlying probabilistic structure. For k = 1, natural structural parameters, such as the position and value of distinguished cells, converge, after rescaling, to Beta(2,1), Beta(1,2), and Uniform (i.e., Beta(1,1)) distributions. These limit laws arise from a coalescence of singularities at the dominant square-root singularity, producing a non-analytic transition in the local expansion. Overall, our results provide both combinatorial insight and a unified analytic perspective on the asymptotic behavior of tree-child networks, showing how algebraic generating functions with interacting singularities systematically produce Beta limit laws.

Tree-child networks are one of the most prominent network classes for modeling evolutionary processes which contain reticulation events. Several recent studies have addressed counting questions for bicombining tree-child networks which are tree-child networks with every reticulation node having exactly two parents. In this paper, we extend these studies to d-combining tree-child networks where every reticulation node has now d ≥ 2 parents. Moreover, we also give results and conjectures on the distributional behavior of the number of reticulation nodes of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves.

We reformulate the Wasserstein Discriminant Analysis (WDA) as a ratio trace problem and present an eigensolver-based algorithm to compute the discriminative subspace of WDA. This new formulation, along with the proposed algorithm, can be served as an efficient and more stable alternative to the original trace ratio formulation and its gradient-based algorithm. We provide a rigorous convergence analysis for the proposed algorithm under the self-consistent field framework, which is crucial but missing in the literature. As an application, we combine WDA with low-dimensional clustering techniques, such as K-means, to perform subspace clustering. Numerical experiments on real datasets show promising results of the ratio trace formulation of WDA in both classification and clustering tasks.