Limiting Probabilities for Vertices of a Given Rank in 1-2 Trees

Bóna, Miklós; Mező, István

doi:10.37236/7731

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…They also studied ternary trees from this aspect [12]. In [7], the first author and István Mező proved the existence of the limits c k for non-plane 1-2 trees and plane 1-2-trees.…”

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confidence: 99%

On a random search tree: asymptotic enumeration of vertices by distance from leaves

Bóna

Pittel

2017

Adv. Appl. Probab.

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A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction c k of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. Notoriously hard to compute, the exact fractions c k had been determined for k ≤ 3 only. We computed c4 and c5 as well; both are ratios of enormous integers, denominator of c5 being 274 digits long. Prompted by the data, we proved that, in sharp contrast, the largest prime divisor of c k 's denominator is 2 k+1 + 1 at most. We conjecture that, in fact, the prime divisors of every denominator for k > 1 form a single interval, from 2 to the largest prime not exceeding 2 k+1 + 1.

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“…They also studied ternary trees from this aspect [12]. In [7], the first author and István Mező proved the existence of the limits c k for non-plane 1-2 trees and plane 1-2-trees.…”

mentioning

confidence: 99%

On a random search tree: asymptotic enumeration of vertices by distance from leaves

Bóna

Pittel

2017

Adv. Appl. Probab.

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“…We note that the asymptotic for the mean follows from the asymptotic for the mean number of leaves which was computed recently in [5] (cf. [4,7,6]), where the same class of trees was referred to as 1-2 trees. By a standard application of Chebyshev's inequality, we see that Theorem 1.1 implies that the number of nodes of outdegree two is concentrated about its mean.…”

Section: Random Lemniscate Treesmentioning

confidence: 99%

The lemniscate tree of a random polynomial

Epstein

Hanin

Lundberg

2021

Annales De l'Institut Fourier

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“…They also studied ternary trees from this aspect [13]. In [7], the first author and István Mező proved the existence of the limits

{c}_k

for nonplane 1‐2 trees and plane 1‐2‐trees.…”

Section: Introductionmentioning

confidence: 99%

Random plane increasing trees: Asymptotic enumeration of vertices by distance from leaves

Bóna

Pittel

2022

Random Struct Algorithms

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We prove that for any fixed k, the probability that a random vertex of a random increasing plane tree is of rank k, that is, the probability that a random vertex is at distance k from the leaves, converges to a constant c k as the size n of the tree goes to infinity. We prove that 1− ∑ j≤k c k < 2 2k+3 (2k+4)! , so that the tail of the limiting rank distribution is super-exponentially narrow. We prove that the latter property holds uniformly for all finite n as well. More generally, we prove that the ranks of a finite uniformly random set of vertices are asymptotically independent, each with distribution {c k }. We compute the exact value of c k for 0 ≤ k ≤ 3, demonstrating that the limiting expected fraction of vertices with rank ≤ 3 is 0.9997… We show that with probability 1 − n −0.99𝜀 the highest rank of a vertex in the tree is sandwiched between (1 − 𝜀) log n∕ log log n and (1.5 + 𝜀) log n∕ log log n, and that this rank is asymptotic to log n∕ log log n with probability 1 − o(1).

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Limiting Probabilities for Vertices of a Given Rank in 1-2 Trees

Abstract: We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree size goes to infinity.

Cited by 3 publications

References 9 publications

On a random search tree: asymptotic enumeration of vertices by distance from leaves

On a random search tree: asymptotic enumeration of vertices by distance from leaves

The lemniscate tree of a random polynomial

Random plane increasing trees: Asymptotic enumeration of vertices by distance from leaves

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