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Fringe subtrees of split trees and fractional split trees

Cecilia Holmgren, Jasper Ischebeck, Svante Janson

Abstract

We consider additive functionals $X_n(φ)$ with small toll functions on split trees and a generalization of split trees, which we call fractional split trees, where the split vector does not need to sum up to 1. These additive functionals encompass e.g. the number of nodes, number of leaves and the number of fringe trees of a certain size. We show convergence of the first moment to a limit, which we can explicitly compute if $s_0=s_1=0$ and for some models with Beta-distributed splitter. For $s_0+s_1>0$, the first moment is given in terms of negative moments of a perpetuity and can often be approximated to arbitrary precision with known bounds. In split trees and certain fractional split trees, the standard deviation is of smaller order than the first moment, where we show a weak law of large numbers. In other fractional split trees, the standard deviation is of the same order and we show a distribution limit using the contraction method.

Fringe subtrees of split trees and fractional split trees

Abstract

We consider additive functionals with small toll functions on split trees and a generalization of split trees, which we call fractional split trees, where the split vector does not need to sum up to 1. These additive functionals encompass e.g. the number of nodes, number of leaves and the number of fringe trees of a certain size. We show convergence of the first moment to a limit, which we can explicitly compute if and for some models with Beta-distributed splitter. For , the first moment is given in terms of negative moments of a perpetuity and can often be approximated to arbitrary precision with known bounds. In split trees and certain fractional split trees, the standard deviation is of smaller order than the first moment, where we show a weak law of large numbers. In other fractional split trees, the standard deviation is of the same order and we show a distribution limit using the contraction method.
Paper Structure (30 sections, 23 theorems, 253 equations, 1 figure)

This paper contains 30 sections, 23 theorems, 253 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. Examples
  4. Toll functions
  5. Nodes, leaves and internal nodes
  6. Balls remaining in the tree
  7. k-protected nodes
  8. Outdegree
  9. Shape functional
  10. Number of subtrees
  11. Independence number, domination number and similar functionals
  12. Generalized Quicksort
  13. Examples of fractional fringe trees
  14. Number of maxima
  15. Partial Search in Quadtrees
  16. ...and 15 more sections

Key Result

Theorem 2.1

Let $\phi$ be a toll function such that $\phi_n = \Ok{n^{\beta-\delta}}$ for some $\delta>0$. This is e.g. true for counting leaves, for counting nodes and for counting fringe subtrees of a certain size. Then, there exists a bounded, continuous, $d$-periodic function $\varpi := \varpi[\phi]$, such t In the nonlattice case, $\varpi$ is constant. If $\phi_n ≥ 0$ for all $n$ and $\phi_n > 0$ for some

Figures (1)

  • Figure 1: The three areas in which the point with maximal sum splits up the triangle.

Theorems & Definitions (62)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3: Weak law of large numbers
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 52 more