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Oscillations in the height of the Yule tree and application to the binary search tree
Author(s) -
Corre PierreAntoine
Publication year - 2017
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20701
Subject(s) - random binary tree , mathematics , binary tree , struct , random walk , random tree , branching (polymer chemistry) , tree (set theory) , branching process , combinatorics , self balancing binary search tree , binary number , statistical physics , discrete mathematics , interval tree , statistics , physics , tree structure , computer science , arithmetic , materials science , motion planning , artificial intelligence , robot , composite material , programming language
For a particular case of a branching random walk with lattice support, namely the Yule branching random walk, we prove that the distribution of the centred maximum oscillates around a distribution corresponding to a critical travelling wave in the following sense: there exist continuous functions t ↦ a tand x ↦ ϕ ¯ ( x ) such that:lim t → + ∞sup x ∈ ℝ | ℙ ( X ¯ ( t ) ≤ a t + x ) − ϕ ¯ ( x − { a t + x } ) | = 0 , where { x } = x − ⌊ x ⌋andX ¯ ( t ) is the height of the Yule tree. We also shows that similar oscillations occur for E ( f ( X ¯ ( t ) − a t ) ) , when f is in a large class of functions. This process is classically related to the binary search tree, thus yielding analogous results for the height and for the saturation level of the binary search tree. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 90–120, 2017