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 $\mathrm{Beta}(2,1)$, $\mathrm{Beta}(1,2)$, and Uniform (i.e., $\mathrm{Beta}(1,1)$) 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.

翻译：树-子网络是一类用于模拟网状进化过程的重要系统发育网络，近年来受到组合学与算法领域研究者的广泛关注。庞斯和巴特提出的一个基本开放问题断言：具有n个叶片和k个网状节点的双组合树-子网络的数量TC_{n,k}等于特定约束词（现称为庞斯-巴特词）的计数。本文证实了该猜想在网状节点数量有界时的情形。我们采用组合分析与解析方法，引入带墙与孔的杨表格族，并构建其与庞斯-巴特词的显式双射，从而给出该恒等式的直接组合解释。这些杨表格编码了底层网络的结构特征，包括网状节点的位置。通过将其投影至带装饰的戴克路径，我们获得带微分算子的代数生成函数（该算子编码步长权重），进而导出TC_{n,k}的显式递推关系与闭式公式。除中等k值的有限检验外，该框架揭示了潜在的随机结构：当k=1时，区分单元的位置与取值等自然结构参数在尺度变换后分别收敛至Beta(2,1)、Beta(1,2)与均匀（即Beta(1,1)）分布。这些极限律源于主导平方根奇点处的奇点融合现象，导致局部展开中出现非解析过渡。综上，我们的结果为树-子网络的渐近行为提供了组合学洞察与统一的解析视角，揭示了代数生成函数中相互作用奇点如何系统性地产生Beta分布极限律。