Total Path Length for Random Recursive Trees

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp signchanges when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

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“…This result is already known; see Mahmoud (1991) and Dobrow and Fill (1999). However, our approach also gives…”

Section: Total Path Lengthsupporting

confidence: 75%

Profiles of random trees: correlation and width of random recursive trees and binary search trees

Drmota

Hwang

2005

Adv. Appl. Probab.

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“…For this tree the limit law for X n was proved by a similar method in Dobrow and Fill [6]. In this paper the authors also derive explicitly the higher moments of the limiting distribution in terms of the ζ-function.…”

Section: Extension To a General Split Tree Modelmentioning

confidence: 89%

On the internal path length ofd-dimensional quad trees

Neininger

Rüschendorf

1999

Random Struct. Alg.

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It is proved that the internal path length of a d-dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the m-ary search trees.

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“…The complexity of the expressions and computations for the first two moments make it seem rather hopeless to compute all the other moments. Nevertheless the first two moments resemble those of the path length which is not normal, see [DF99].…”

Section: Recursive Treesmentioning

confidence: 97%

On the shape of the fringe of various types of random trees

Drmota

Gittenberger

Panholzer

et al. 2008

Math Methods in App Sciences

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Abstract. We analyze a fringe tree parameter w in a variety of settings, utilizing a variety of methods from the analysis of algorithms and data structures.Given a tree t and one of its leaves a, the w(t, a) parameter denotes the number of internal nodes in the subtree rooted at a's father. The closely-related w(t, a) parameter denotes the number of leaves, excluding a, in the subtree rooted at a's father. We define the cumulative w parameter as W(t) = a w(t, a), i.e., as the sum of w(t, a) over all leaves a of t. The w parameter not only plays an important rôle in the analysis of the Lempel-Ziv '77 data compression algorithm, but it is captivating from a combinatorial viewpoint too.In this report, we determine the asymptotic behavior of the w and W parameters on a variety of types of trees. In particular, we analyze simply generated trees, recursive trees, binary search trees, digital search trees, tries and Patricia tries.The final section of this report briefly summarizes and improves the previously known results about the w parameter's behavior on tries and suffix trees, originally published in one author's thesis (see [War05]This survey of new results about the w parameter is very instructive since a variety of different combinatorial methods are used in tandem to carry out the analysis.

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Total Path Length for Random Recursive Trees

Cited by 39 publications

References 21 publications

Profiles of random trees: correlation and width of random recursive trees and binary search trees

Profiles of random trees: correlation and width of random recursive trees and binary search trees

On the internal path length ofd-dimensional quad trees

On the shape of the fringe of various types of random trees

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