AI 中文总结
研究条件为有\(n\)个顶点和\(k_n\)个叶子（\(k_n = o(n)\)）的比内-高尔顿-沃森树，通过卢卡西维茨路径编码及随机游走渐近估计，确定其出度渐近分布，明确不同出度顶点出现的临界尺度及相关规律。
AI 中文摘要
我们研究了条件为具有\(n\)个顶点和\(k_n\)个叶子的比内-高尔顿-沃森树，其中\(k_n\to\infty\)且相对于\(n\)可忽略不计，即\(k_n = o(n)\)。更确切地说，我们确定了这些树的出度的渐近分布。首先表明树渐近为二叉树：有两个孩子的顶点数量渐近等于叶子数量，而几乎所有其余顶点恰好有一个孩子。然后，我们确定了具有更大出度的顶点出现的尺度。对于每个\(d\geq2\)，临界尺度\(k_n\sim cn^{(d - 1)/d}\)，\(c>0\)，是出度为\(d + 1\)的顶点出现的阈值。低于此尺度，此类顶点大概率不存在；在临界尺度，其数量收敛到泊松分布；高于此尺度，它们满足大数定律。我们的证明依赖于通过卢卡西维茨路径对比内-高尔顿-沃森树的编码以及相关随机游走的渐近估计。
英文摘要
We study Bienaymé-Galton-Watson trees conditioned to have $n$ vertices and $k_n$ leaves, where $k_n\to\infty$ while remaining negligible compared to $n$, i.e. $k_n=o(n)$. More precisely, we determine the asymptotic distribution of the outdegrees of these trees. We first show that the tree is asymptotically binary: the number of vertices with two children is asymptotically equal to the number of leaves, while almost all remaining vertices have exactly one child. Then, we identify the scales at which vertices with larger outdegrees emerge. For every $d\ge2$, the critical scale $k_n \sim cn^{(d-1)/d}$, $c>0$, is the threshold for the appearance of vertices with outdegree $d+1$. Below this scale, such vertices are absent with high probability; at the critical scale, their number converges to a Poisson distribution; above it, they satisfy a law of large numbers. Our proofs rely on the coding of Bienaymé-Galton-Watson trees by their Łukasiewicz paths and asymptotic estimates for associated random walks.
