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Sackin Indices for Labeled and Unlabeled Classes of Galled Trees

Michael Fuchs, Bernhard Gittenberger

TL;DR

This work extends the Sackin balance index to the class of galled trees and its simplex and normal subclasses, for both labeled and unlabeled networks. Employing the symbolic method of analytic combinatorics, the authors derive exact asymptotics for the Sackin index means, establish moment expansions, and prove that the properly scaled Sackin indices converge in distribution (and in all moments) to the Airy distribution. The results reveal that the mean Sackin index grows like $\mu n^{3/2}$ with class-dependent constants $\mu$, and that the Airy distribution governs the limiting behavior across all considered variants. The methods also yield precise counting information for these network classes and pave the way for further balance indices in phylogenetic networks, including potential extensions to $B_2$.

Abstract

The Sackin index is an important measure for the balance of phylogenetic trees. We investigate two extensions of the Sackin index to the class of galled trees and two of its subclasses (simplex galled trees and normal galled trees) where we consider both labeled and unlabeled galled trees. In all cases, we show that the mean of the Sackin index for a network which is uniformly sampled from its class is asymptotic to $μn^{3/2}$ for an explicit constant $μ$. In addition, we show that the scaled Sackin index convergences weakly and with all its moments to the Airy distribution.

Sackin Indices for Labeled and Unlabeled Classes of Galled Trees

TL;DR

This work extends the Sackin balance index to the class of galled trees and its simplex and normal subclasses, for both labeled and unlabeled networks. Employing the symbolic method of analytic combinatorics, the authors derive exact asymptotics for the Sackin index means, establish moment expansions, and prove that the properly scaled Sackin indices converge in distribution (and in all moments) to the Airy distribution. The results reveal that the mean Sackin index grows like with class-dependent constants , and that the Airy distribution governs the limiting behavior across all considered variants. The methods also yield precise counting information for these network classes and pave the way for further balance indices in phylogenetic networks, including potential extensions to .

Abstract

The Sackin index is an important measure for the balance of phylogenetic trees. We investigate two extensions of the Sackin index to the class of galled trees and two of its subclasses (simplex galled trees and normal galled trees) where we consider both labeled and unlabeled galled trees. In all cases, we show that the mean of the Sackin index for a network which is uniformly sampled from its class is asymptotic to for an explicit constant . In addition, we show that the scaled Sackin index convergences weakly and with all its moments to the Airy distribution.
Paper Structure (16 sections, 5 theorems, 117 equations, 5 figures, 1 table)

This paper contains 16 sections, 5 theorems, 117 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction and Results
  2. Symbolic Method and Generating Functions
  3. Labeled structures and generating functions
  4. Unlabeled structures
  5. The generating function for galled trees
  6. Getting the coefficients
  7. The Mean of the Sackin Index in Galled Trees
  8. Limit Law of Sackin Indices
  9. Variations
  10. Simplex galled trees
  11. Normal galled trees
  12. Unlabeled Galled Trees
  13. General case
  14. Simplex galled trees
  15. Normal galled trees
  16. ...and 1 more sections

Key Result

Theorem 1

For all cases, as $n\rightarrow\infty$, where $\mu^{(*,**)}$ with $*\in\{\ell,u\}$ and $**\in\{\max,\min\}$ is given in Table mean_const and Table mean_const_unlab.

Figures (5)

  • Figure 1: Four unlabeled galled trees, each with exactly two galls. Upper left: A normal galled tree that is not simplex, as the child of the red reticulation node is not a leaf. Upper right: A simplex and normal galled tree. Lower left: A galled tree that is not normal, because the end points of the red edge are in a ancestor-descendant relationship. It is not simplex either because of the child of the red reticulation node. Lower right: A simplex, but not normal galled tree.
  • Figure 2: Symbolic specification of galled trees. In the last of these four cases, we assume that the structure is drawn in such a way that $k\ge \ell$.
  • Figure 3: The domain where we require analyticity.
  • Figure 4: Symbolic equation for simplex galled trees. The red substructures are the places where galled trees and simplex galled trees differ.
  • Figure 5: Symbolic specification of normal galled trees.

Theorems & Definitions (26)

  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Remark 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Theorem 2
  • ...and 16 more