The Kingman coalescent process is a classical model of gene genealogies in population genetics. It generates Yule-distributed, binary ranked tree topologies – also called histories – with a finite number of $n$ leaves, together with $n-1$ exponentially distributed time lengths: one for each layer of the history. Using a discrete approach, we study the lengths of the external branches of Yule distributed histories, where the length of an external branch is defined as the rank of its parent node. We study the multiplicity of external branches of given length in a random history of $n$ leaves. A correspondence between the external branches of the ordered histories of size $n$ and the non-peak entries of the permutations of size $n-1$ provides easy access to the length distributions of the first and second longest external branches in a random Yule history and coalescent tree of size $n$. The length of the longest external branch is also studied in dependence of root balance of a random tree. As a practical application, we compare the observed and expected number of mutations on the longest external branches in samples from natural populations.

金曼合并过程是群体遗传学中基因谱系的经典模型。它会生成有限的数量的Yule分布式，二进制排名的树形拓扑（也称为历史）。$ñ$ 叶子，连同 $ñ-\mathrm{1个}$指数分布的时间长度：历史记录的每一层都有一个。使用离散方法，我们研究了Yule分布式历史的外部分支的长度，其中外部分支的长度定义为其父节点的等级。我们研究随机历史中给定长度的外部分支的多样性$ñ$树叶。有序历史大小的外部分支之间的对应关系$ñ$ 以及大小排列的非峰值条目 $ñ-\mathrm{1个}$ 在随机的Yule历史和大小合并的树中，可以轻松访问第一和第二最长外部分支的长度分布 $ñ$。还根据随机树的根平衡来研究最长的外部分支的长度。作为实际应用，我们比较了自然种群样本中最长的外部分支上观察到的突变和预期的突变数量。