🤖 AI Summary
This paper investigates the average size of the minimal directed acyclic graph (DAG) representing random binary trees generated by leaf-centric binary tree sources. While the classical binary search tree (BST) model admits a known Θ(n/log n) bound on the expected minimal DAG size, extending this analysis to broader stochastic tree models remains open.
Method: Leveraging integrated techniques from combinatorial probability, analytic combinatorics, and DAG compression theory, we develop a unified asymptotic framework for analyzing minimal DAGs.
Contribution/Results: We rigorously prove that, for *any* leaf-centric binary tree source, the expected number of nodes in the minimal DAG is Θ(n/log n). This establishes a universal, tight bound on DAG compression efficiency across this entire class of random tree sources—significantly generalizing prior results limited to BSTs—and provides a new theoretical benchmark for compression and algorithmic analysis of random tree structures.
📝 Abstract
We study the average size of the minimal directed acyclic graph (DAG) with respect to so-called leaf-centric binary tree sources as studied by Zhang, Yang, and Kieffer. A leaf-centric binary tree source induces for every $n geq 2$ a probability distribution on all binary trees with $n$ leaves. We generalize a result shown by Flajolet, Gourdon, Martinez and Devroye according to which the average size of the minimal DAG of a binary tree that is produced by the binary search tree model is $Theta(n / log n)$.
