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.

The purpose of this paper is to provide a central limit theorem for the number of occurrences of double triangles in random planar maps. This is the first result of this kind that goes beyond face counts of given valency. The method is based on generating functions, an involved combinatorial decomposition scheme that leads to a system of catalytic functional equations and an analytic extension of the Quadratic Method to systems of equations.