Depth of nodes in random recursive k-ary trees

Javanian, Mehri; Vahidi-Asl, Mohammad Q.

doi:10.1016/j.ipl.2005.11.020

Cited by 8 publications

(8 citation statements)

References 5 publications

(3 reference statements)

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“…Remark 4.1. Lemma 4.2 can also be proved by a direct calculation using an exact formula for the expectation of the unweighted depth given in Javanian and Vahidi-Asl (2006). We then obtain the constant c p in (4.4) in terms of an infinite series.…”

Section: The Internal Path Length and The Wiener Indexmentioning

confidence: 77%

“…Thus, the claims for the expectation and variance in (3.1) follow from the results of Javanian and Vahidi-Asl (2006) forD n . Now, let…”

Section: Limit Theorems For Depths and Distancesmentioning

confidence: 82%

“…We then use this result to derive the central limit theorem of a randomly chosen node D U in Corollary 3.1 below. The result of Javanian and Vahidi-Asl (2006) corresponds to the case of all edge weights being 1, i.e. µ = 1 and σ 2 = 0.…”

Section: Limit Theorems For Depths and Distancesmentioning

confidence: 99%

“…The expectation of the internal path length is given in Bergeron et al (1992) for the unweighted tree. It can also be obtained by summing up the exact expectations for the unweighted depths given in Javanian and Vahidi-Asl (2006).…”

Section: The Internal Path Length and The Wiener Indexmentioning

confidence: 99%

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Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

Munsonius

Rüschendorf

2011

Journal of Applied Probability

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Limit theorems are established for some functionals of the distances between two nodes in weighted random b-ary recursive trees. We consider the depth of the nth node and of a random node, the distance between two random nodes, the internal path length, and the Wiener index. As an application, these limit results imply, by an imbedding argument, corresponding limit theorems for further classes of random trees: plane-oriented recursive trees and random linear recursive trees.

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Section: The Internal Path Length and The Wiener Indexmentioning

confidence: 77%

“…Thus, the claims for the expectation and variance in (3.1) follow from the results of Javanian and Vahidi-Asl (2006) forD n . Now, let…”

Section: Limit Theorems For Depths and Distancesmentioning

confidence: 82%

Section: Limit Theorems For Depths and Distancesmentioning

confidence: 99%

Section: The Internal Path Length and The Wiener Indexmentioning

confidence: 99%

See 2 more Smart Citations

Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

Munsonius

Rüschendorf

2011

Journal of Applied Probability

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“…The data structure was composed by three main object classes, corresponding to the MTG scales. The recursive routing algorithm (Javanian and Vahidi-Asl 2006) was applied to convert the CoffeePlant3D data structure to MTG or vice versa. This algorithm routed all metamer objects, starting from the orthotropic axis, and writing all metamers respecting the hierarchy of branching orders.…”

Section: Resultsmentioning

confidence: 99%

Strategies to reconstruct 3D Coffea arabica L. plant structure

et al. 2016

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Accurate model of structural elements is necessary to model the foliage and fruit distributions in cultivated plants, both of them being key parameters for yield prediction. However, the level of details in architectural data collection could vary, simplifying the data collection when plants get older and because of the high time cost required. In the present study, we aimed at reconstructing and analyzing plant structure, berry distributions and yield in Coffea arabica (Arabica coffee), by using both detailed or partial morphological information and probabilistic functions. Different datasets of coffee plant architectures were available with different levels of detail depending on the tree age. Three scales of decomposition—plant, axes and metamers were used reconstruct the plant architectures. CoffePlant3D, a software which integrates a series of mathematical, computational and statistical methods organized in three newly developed modules, AmostraCafe3D, VirtualCafe3D and Cafe3D, was developed to accurately reconstruct coffee plants in 3D, whatever the level of details available. The number of metamers of the 2nd order axes was shown to be linearly proportional to that of the orthotropic trunk, and the number of berries per metamer was modeled as a Gaussian function within a specific zone along the plagiotropic axes. This ratio of metamer emission rhythm between the orthotropic trunk and plagiotropic axes represents the pillar of botanical events in the C. arabica development and was central in our modeling approach, especially to reconstruct missing data. The methodology proposed for reconstructing coffee plants under the CoffePlant3D was satisfactorily validated across dataset available and could be performed for any other Arabica coffee variety.

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Profile of Random Exponential Recursive Trees

Mahmoud

2021

Methodol Comput Appl Probab

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Depth of nodes in random recursive k-ary trees

Cited by 8 publications

References 5 publications

Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

Strategies to reconstruct 3D Coffea arabica L. plant structure

Profile of Random Exponential Recursive Trees

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