The decompressed tree size of $k$-ary chains
Provides theoretical analysis for tree compression, relevant to researchers in data compression and combinatorics.
The paper computes the asymptotic expected decompressed tree size of a uniformly random k-ary chain of size n, revealing a stretched exponential term e^{c√n}. This result also impacts the limit distribution of Brauer chains of fixed length.
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.