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Asymptotics of relaxed $k$-ary trees

Manosij Ghosh Dastidar, Michael Wallner

TL;DR

This work establishes a unified asymptotic framework for relaxed $k$-ary trees, showing that their counts exhibit a stretched exponential factor in $n$ for all fixed arities $k\ge 2$. By transforming the combinatorial recurrence into a Dyck-like two-step process and performing Airy-function–driven analysis, the authors derive tight $\Theta$-bounds that incorporate a universal Airy constant $a_1$ and a $n^{(2k-1)/3}$ polynomial term. The approach combines bijections to decorated paths, a careful rescaling, heuristic Airy-based ansatz, and rigorous upper/lower bounds via lattice-path arguments, culminating in a complete proof for the relaxed $k$-ary case and suggesting broader applicability to related DAG classes. These results deepen understanding of substructure compression phenomena and have potential implications for analyzing compacted trees and minimal deterministic finite automata over finite alphabets. Practically, the findings explain repeated-substructure compression limits and inform the asymptotic behavior of associated counting sequences.

Abstract

A relaxed $k$-ary tree is an ordered directed acyclic graph with a unique source and sink in which every node has out-degree $k$. These objects arise in the compression of trees in which some repeated subtrees are factored and repeated appearances are replaced by pointers. We prove an asymptotic theta-result for the number of relaxed $k$-ary tree with $n$ nodes for $n \to \infty$. This generalizes the previously proved binary case to arbitrary finite arity, and shows that the seldom observed phenomenon of a stretched exponential term $e^{c n^{1/3}}$ appears in all these cases. We also derive the recurrences for compacted $k$-ary trees in which all subtrees are unique and minimal deterministic finite automata accepting a finite language over a finite alphabet.

Asymptotics of relaxed $k$-ary trees

TL;DR

This work establishes a unified asymptotic framework for relaxed -ary trees, showing that their counts exhibit a stretched exponential factor in for all fixed arities . By transforming the combinatorial recurrence into a Dyck-like two-step process and performing Airy-function–driven analysis, the authors derive tight -bounds that incorporate a universal Airy constant and a polynomial term. The approach combines bijections to decorated paths, a careful rescaling, heuristic Airy-based ansatz, and rigorous upper/lower bounds via lattice-path arguments, culminating in a complete proof for the relaxed -ary case and suggesting broader applicability to related DAG classes. These results deepen understanding of substructure compression phenomena and have potential implications for analyzing compacted trees and minimal deterministic finite automata over finite alphabets. Practically, the findings explain repeated-substructure compression limits and inform the asymptotic behavior of associated counting sequences.

Abstract

A relaxed -ary tree is an ordered directed acyclic graph with a unique source and sink in which every node has out-degree . These objects arise in the compression of trees in which some repeated subtrees are factored and repeated appearances are replaced by pointers. We prove an asymptotic theta-result for the number of relaxed -ary tree with nodes for . This generalizes the previously proved binary case to arbitrary finite arity, and shows that the seldom observed phenomenon of a stretched exponential term appears in all these cases. We also derive the recurrences for compacted -ary trees in which all subtrees are unique and minimal deterministic finite automata accepting a finite language over a finite alphabet.
Paper Structure (8 sections, 10 theorems, 33 equations, 3 figures, 3 tables)

This paper contains 8 sections, 10 theorems, 33 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction and Main Result
  2. Bijections and recurrences
  3. Asymptotics of relaxed k-ary trees
  4. Transformation into a Dyck-like recurrence
  5. Heuristic analysis
  6. Explicit bounds
  7. Proof of Theorem \ref{['theo:relaxedasymptotics']} on relaxed k-ary trees
  8. Conclusion and Outlook

Key Result

Theorem 1

Let $k\geq2$ be an integer. The number of relaxed $k$-ary trees satisfies for $n \to \infty$ where $a_1 \approx -2.338$ is the largest root of the Airy function $\text{\normalfont Ai}(x)$ defined as the unique function satisfying $\text{\normalfont Ai}"(x)=x \text{\normalfont Ai}(x)$ and $\lim_{x \to \infty} \text{\normalfont Ai}(x)=0$.

Figures (3)

  • Figure 1: All $7$ ternary relaxed ternary trees with $2$ internal nodes.
  • Figure 2: Example of relaxed ternary tree with 7 internal nodes (circles) labelled in postorder. The unique sink is depicted by a square. The black edges belong to the spine, the red ones are so-called pointers.
  • Figure 3: Example of the bijection described in Lemma \ref{['lem:bijectionrelaxed']} between relaxed ternary trees and horizontally $3$-decorated paths.

Theorems & Definitions (14)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 6
  • Proposition 7
  • Theorem 8
  • Proposition 9
  • Proposition 10
  • ...and 4 more