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On the enumeration of records of rooted trees and rooted forests

Adrián Lillo, Mercedes Rosas, Stefan Trandafir

TL;DR

This work provides a systematic enumeration of records in rooted Cayley trees and rooted forests by introducing a record decomposition (bonsai sequence and attachment data) and connecting the associated generating functions to the Cayley tree function $\mathcal{T}(z)$. It yields closed formulas for tree- and forest-record numbers, notably $R_{\bullet}(n,k)=k(n-1)\cdots(n-k+1)n^{n-k-1}$ and a Stirling-number expression for forests, and derives functional equations linking these generating functions to $\mathcal{T}(z)$. The authors establish recurrences, asymptotics, and structural properties such as log-concavity and peak location, linking combinatorics of records to broader topics like parking functions and queueing theory. The results illuminate the refined distribution of records and provide tools for further analytic and bijective explorations in labelled tree/forest combinatorics, with practical implications for related stochastic models.

Abstract

A record of a rooted Cayley tree is a node whose label is the largest along the unique path to the root. In this work, we find elegant functional equations relating the generating functions for records of rooted Cayley trees and for records of forests of rooted trees with the Cayley tree function, and explore the consequences of our results.

On the enumeration of records of rooted trees and rooted forests

TL;DR

This work provides a systematic enumeration of records in rooted Cayley trees and rooted forests by introducing a record decomposition (bonsai sequence and attachment data) and connecting the associated generating functions to the Cayley tree function . It yields closed formulas for tree- and forest-record numbers, notably and a Stirling-number expression for forests, and derives functional equations linking these generating functions to . The authors establish recurrences, asymptotics, and structural properties such as log-concavity and peak location, linking combinatorics of records to broader topics like parking functions and queueing theory. The results illuminate the refined distribution of records and provide tools for further analytic and bijective explorations in labelled tree/forest combinatorics, with practical implications for related stochastic models.

Abstract

A record of a rooted Cayley tree is a node whose label is the largest along the unique path to the root. In this work, we find elegant functional equations relating the generating functions for records of rooted Cayley trees and for records of forests of rooted trees with the Cayley tree function, and explore the consequences of our results.
Paper Structure (11 sections, 21 theorems, 62 equations, 5 figures, 3 tables)

This paper contains 11 sections, 21 theorems, 62 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Records, record decomposition, and record numbers.
  3. The record decomposition
  4. Some enumerative consequences of the record decomposition.
  5. The Cayley tree function, and record generating functions for rooted trees and rooted forests
  6. Record numbers
  7. Consequences
  8. Asymptotics
  9. Log-concavity
  10. Asymptotic peak location
  11. Final Comments

Key Result

Lemma 2.3

Let $t=(t_1, t_2, \ldots, t_k)$ be a composition of $n$. The number of restricted flags of type $t$ is given by

Figures (5)

  • Figure 1: All 16 trees with 3 vertices rooted at $\circ$. Record nodes are drawn in black. Nine of these trees are planted trees.
  • Figure 2: A tree $T$ labelled with $[14]_0$ and rooted at $\circ$. The record nodes of $T$ are drawn in black. The root $\circ$ is not considered a record node.
  • Figure 3: The tree $T$ with edges above record nodes deleted.
  • Figure 4: The record decomposition of $T$ has bonsai type $(4,1,1,2,5,1) \vDash 14$, attachment sequence $(\circ, 1, 4, \circ, 11),$ and bonsai sequence as illustrated.
  • Figure 5: The recursive decomposition of Lemma \ref{['le:recurrence_forests']}.

Theorems & Definitions (41)

  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 31 more