The Decompressed Tree Size of k-Ary Chains

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 ≥ 2 k ≥ 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 ec \, √n e c n . This result also has implications for the limit distribution of Brauer chains of fixed length.