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Fringe trees of Patricia tries, compressed binary search trees, and three other random full binary trees

Svante Janson

TL;DR

The paper analyzes fringe-tree distributions across several random full binary-tree models, focusing on counts of fixed fringe trees and their asymptotic behavior. It develops central limit theorems and distinguishes quenched and annealed limits, with Patricia tries exhibiting periodic oscillations and extended fringe trees enabling unified limit descriptions. It covers extended and compressed binary search trees, along with the uniform full binary tree and the beta-splitting family (including the critical case), providing explicit formulas, recursions, and generating-function methods to compute fringe-distribution constants. The results build on uncompressed-tree theory and offer practical computation techniques for fringe statistics, including Mellin-transform based formulas and recursive schemes for the compressed BST case, with comparisons to real cladogram data.

Abstract

We study the distribution of fringe trees in Patricia tries (extending earlier results by Ischebeck (2025)) and compressed binary search trees; both cases are random binary trees that have been compressed by deleting nodes of outdegree 1 so that they are random full binary trees. The main results are central limit theorems for the number of fringe trees of a given type, which imply quenched and annealed limit results for the fringe tree distribution; for Patricia tries, this is complicated by periodic oscillations in the usual manner. We also consider extended fringe trees. The results are derived from earlier results for uncompressed tries and binary search trees. In the case of compressed binary search trees, it seems difficult to give a closed formula for the asymptotic fringe tree distribution, but we provide a recursion and give examples. For comparison, we give also results, simpler and partly known, for three other models of random full binary trees: the extended binary search tree, the critical beta-spltting random tree, and the uniform random full binary tree.

Fringe trees of Patricia tries, compressed binary search trees, and three other random full binary trees

TL;DR

The paper analyzes fringe-tree distributions across several random full binary-tree models, focusing on counts of fixed fringe trees and their asymptotic behavior. It develops central limit theorems and distinguishes quenched and annealed limits, with Patricia tries exhibiting periodic oscillations and extended fringe trees enabling unified limit descriptions. It covers extended and compressed binary search trees, along with the uniform full binary tree and the beta-splitting family (including the critical case), providing explicit formulas, recursions, and generating-function methods to compute fringe-distribution constants. The results build on uncompressed-tree theory and offer practical computation techniques for fringe statistics, including Mellin-transform based formulas and recursive schemes for the compressed BST case, with comparisons to real cladogram data.

Abstract

We study the distribution of fringe trees in Patricia tries (extending earlier results by Ischebeck (2025)) and compressed binary search trees; both cases are random binary trees that have been compressed by deleting nodes of outdegree 1 so that they are random full binary trees. The main results are central limit theorems for the number of fringe trees of a given type, which imply quenched and annealed limit results for the fringe tree distribution; for Patricia tries, this is complicated by periodic oscillations in the usual manner. We also consider extended fringe trees. The results are derived from earlier results for uncompressed tries and binary search trees. In the case of compressed binary search trees, it seems difficult to give a closed formula for the asymptotic fringe tree distribution, but we provide a recursion and give examples. For comparison, we give also results, simpler and partly known, for three other models of random full binary trees: the extended binary search tree, the critical beta-spltting random tree, and the uniform random full binary tree.
Paper Structure (21 sections, 22 theorems, 185 equations, 1 figure, 2 tables)

This paper contains 21 sections, 22 theorems, 185 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The studied trees
  3. Tries
  4. Binary search trees
  5. Uniform random full binary trees
  6. Beta-splitting random trees
  7. Fringe trees
  8. Preliminaries
  9. Notation
  10. Tries
  11. Binary search trees
  12. Fringe trees of compressed trees
  13. Extended fringe trees
  14. Patricia tries
  15. Extended binary search trees
  16. ...and 6 more sections

Key Result

Lemma 2.3

Let $(T_n)$ be a sequence of random trees such that, as ${n\to\infty}$, for every fixed tree $t$ and some random tree $T^*_\infty$ (a quenched limiting fringe tree). Then where (with the limit in probability, in general) In other words, the (quenched) distribution of the extended fringe tree converges in probability: where the distribution of the limiting extended fringe tree ${T}^{**}_\infty$

Figures (1)

  • Figure 1: Some small full binary trees.

Theorems & Definitions (66)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1: Partly Ischebeck
  • proof
  • Remark 3.2
  • ...and 56 more