Notes on protected nodes in digital search trees

Du, Rosena R. X.; Prodinger, Helmut

doi:10.1016/j.aml.2011.11.017

Cited by 22 publications

(17 citation statements)

References 8 publications

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“…If v is a vertex of a tree T then let the rank of v be the number of edges in the shortest path from v to a leaf of T that is a descendant of v. So leaves are of rank 0, neighbors of leaves are of rank 1, and so on. Motivated by a series of recent papers [7], [14] concerning the neighbors of leaves, Bóna [2] proved that, for any k ≥ 0, the probability that a randomly selected vertex of a randomly selected tree is of rank k converges to a rational number c k as n goes to ∞. He also computed that c 0 = 1 3 , c 1 = 3 10 , c 2 = 1721 8100 , and c 3 ≈ 0.105.…”

Section: Recent Resultsmentioning

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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Section: Recent Resultsmentioning

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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“…There are many recent studies of so-called protected nodes in various classes of random trees, see e.g. [4,6,11,13,35,36,19,20,21]. A node is protected (more precisely, two-protected) if it is not a leaf and none of its children is a leaf.…”

Section: 2mentioning

confidence: 99%

Multivariate Normal Limit Laws for the Numbers of Fringe Subtrees in $m$-ary Search Trees and Preferential Attachment Trees

Holmgren

Janson

Šileikis

2017

Electron. J. Combin.

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We study fringe subtrees of random m-ary search trees and of preferential attachment trees, by putting them in the context of generalised Pólya urns. In particular we show that for the random m-ary search trees with m ≤ 26 and for the linear preferential attachment trees, the number of fringe subtrees that are isomorphic to an arbitrary fixed tree T converges to a normal distribution; more generally, we also prove multivariate normal distribution results for random vectors of such numbers for different fringe subtrees. Furthermore, we show that the number of protected nodes in random m-ary search trees for m ≤ 26 has asymptotically a normal distribution.

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“…Moreover, this equation holds for n smaller than 4 too, except at n = 3. So, we adjust by z 3 , and we obtain a differential equation for these generating functions:…”

Section: The Variancementioning

confidence: 99%

“…Cheon and Shapiro [2] investigated the average number of protected nodes in unlabeled ordered trees and in Motzkin trees. Mansour [6] considered the average number of protected nodes in k-ary trees, Du and Prodinger [3] analyzed the average number of protected nodes in random digital trees, and Mahmoud and Ward [5] presented a central limit theorem, as well as exact moments of all orders, for the number of protected nodes in binary search trees.…”

Section: Introductionmentioning

confidence: 99%