[Paper Review] Almost sure asymptotics for the random binary search tree

Matthew I. Roberts|arXiv (Cornell University)|Feb 22, 2010

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes almost sure asymptotic fluctuations for the height $ H_n $ and saturation level $ h_n $ of random binary search trees on the $ \log\log $ scale, showing that $ \frac{b\log n - aH_n}{\log\log n} $ oscillates between $ \frac{1}{2} $ and $ \frac{3}{2} $ almost surely, with convergence in probability to $ \frac{3}{2} $. It further proves that the number of particles at the tree's frontier, $ F_n $, is unbounded almost surely, indicating persistent growth in the fringe size.

ABSTRACT

We consider a (random permutation model) binary search tree with n nodes and give asymptotics on the loglog scale for the height H_n and saturation level h_n of the tree as n o\infty, both almost surely and in probability. We then consider the number F_n of particles at level H_n at time n, and show that F_n is unbounded almost surely.

Motivation & Objective

• To derive almost sure asymptotic fluctuations for the height $ H_n $ and saturation level $ h_n $ of random binary search trees on the $ \log\log $ scale.
• To establish precise almost sure liminf and limsup behavior for normalized deviations of $ H_n $ and $ h_n $ from their logarithmic growth.
• To investigate the long-term behavior of the number of particles $ F_n $ at the frontier (level $ H_n $), particularly whether it remains bounded or grows indefinitely.
• To leverage the connection between binary search trees and the continuous-time Yule branching process to transfer results from branching random walks to the discrete tree model.
• To provide a foundation for further analysis of the asymptotic distribution of $ F_n $, which remains an open problem in the current work.

Proposed method

• Utilizes the well-known correspondence between random binary search trees and the Yule branching process in continuous time, where each particle branches at rate 1 and moves left by 1 upon splitting.
• Applies a key result from Hu & Shi (2007) on the almost sure limsup and liminf behavior of the minimum position in a Yule process to derive asymptotics for $ H_n $ and $ h_n $.
• Defines normalized deviations $ \frac{b\log n - aH_n}{\log\log n} $ and $ \frac{\alpha h_n - \beta\log n}{\log\log n} $, with constants $ a, b, \alpha, \beta $ derived from implicit equations related to branching dynamics.
• Uses the strong Markov property and exponential tail estimates to bound the probability that a single lineage reaches the frontier before existing particles branch.
• Employs an inductive argument on the number of frontier particles $ 2k $, showing that if $ 2k $ particles appear infinitely often, then $ 2k+2 $ particles also appear infinitely often with probability 1.
• Transfers results from the continuous-time Yule process to the discrete binary search tree via path equivalence up to time change, enabling almost sure conclusions for $ F_n $.

Experimental results

Research questions

• RQ1 What are the almost sure liminf and limsup fluctuations of the height $ H_n $ of a random binary search tree on the $ \log\log $ scale?
• RQ2 How do the saturation level $ h_n $ and height $ H_n $ deviate from their logarithmic growth rates in the almost sure sense?
• RQ3 Is the number of particles $ F_n $ at the frontier (level $ H_n $) bounded almost surely, or does it grow without bound?
• RQ4 Can the dynamics of particle movement and branching in the Yule process be used to infer properties of the discrete binary search tree?
• RQ5 What is the probability that a lineage from a particle near the frontier reaches the frontier before existing particles branch, and how does this affect the growth of $ F_n $?

Key findings

• The normalized deviation $ \frac{b\log n - aH_n}{\log\log n} $ satisfies $ \frac{1}{2} = \liminf_{n\to\infty} \frac{b\log n - aH_n}{\log\log n} < \limsup_{n\to\infty} \frac{b\log n - aH_n}{\log\log n} = \frac{3}{2} $ almost surely, with $ a \approx 0.76804 $, $ b \approx 3.31107 $.
• In probability, $ \frac{b\log n - aH_n}{\log\log n} \to \frac{3}{2} $ as $ n \to \infty $, confirming convergence in distribution to the limsup value.
• For the saturation level $ h_n $, the normalized deviation $ \frac{\alpha h_n - \beta\log n}{\log\log n} $ has the same liminf and limsup bounds: $ \frac{1}{2} $ and $ \frac{3}{2} $ almost surely, with $ \alpha \approx 1.6783 $, $ \beta \approx 0.6266 $.
• The number of particles $ F_n $ at the frontier satisfies $ \limsup_{n\to\infty} F_n = \infty $ almost surely, indicating that the fringe size grows without bound.
• The proof of unboundedness of $ F_n $ relies on showing that, given $ 2k $ particles at the frontier, there is a positive probability (uniformly bounded away from zero) that two new particles reach the frontier before any of the existing $ 2k $ branch, leading to infinitely many visits to $ 2k+2 $ particles.
• The result is established via induction on $ k $, using the strong Markov property and exponential moment estimates to ensure recurrence of frontier sizes beyond any finite $ 2k $.