A chain is defined as a directed acyclic graph (DAG) with one source and one sink, where the children are ordered and the spanning tree computed using a depth-first search is a path. Such DAGs emerge in the context of tree compression and are therefore uniquely associated with a tree. The tree size of a DAG is defined as the size of the associated tree. For fixed out-degree $k \geq 2$, we compute the asymptotic expected decompressed tree size of a chain of size $n$ chosen uniformly at random, and we show that it contains a stretched exponential term of the form $e^{c \, \sqrt{n}}$. This result also has implications for the limit distribution of Brauer chains of fixed length.

翻译：链被定义为具有一个源点和一个汇点的有向无环图（DAG），其中子节点有序且使用深度优先搜索计算的生成树是一条路径。此类DAG出现在树压缩的背景下，因此与某棵树唯一关联。DAG的树大小定义为关联树的大小。对于固定出度$k \geq 2$，我们计算了均匀随机选取的规模为$n$的链的渐近期望解压缩树大小，并证明其包含一个形如$e^{c \, \sqrt{n}}$的拉伸指数项。该结果还对固定长度的Brauer链的极限分布具有意义。