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Random Recursive Trees and Preferential Attachment Trees are Random Split Trees

Part of: Graph theory Combinatorial probability

Published online by Cambridge University Press:  21 May 2018

SVANTE JANSON
SVANTE JANSON*
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden (e-mail: [email protected], http://www.math.uu.se/svante-janson)
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Abstract

We consider linear preferential attachment trees, and show that they can be regarded as random split trees in the sense of Devroye (1999), although with infinite potential branching. In particular, this applies to the random recursive tree and the standard preferential attachment tree. An application is given to the sum over all pairs of nodes of the common number of ancestors.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C05: Trees
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 1 , January 2019 , pp. 81 - 99
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

Partly supported by the Knut and Alice Wallenberg Foundation.

References

[1] Aldous, D. J. (1985) Exchangeability and related topics. In École d'Été de Probabilités de Saint-Flour XIII, 1983, Vol. 1117 of Lecture Notes in Mathematics, Springer, pp. 1–198.Google Scholar
[2] Athreya, K. B. (1969) On a characteristic property of Polya's urn. Studia Sci. Math. Hungar. 4 31–35.Google Scholar
[3] Barabási, A.-L. and Albert, R. (1999) Emergence of scaling in random networks. Science 286 (5439), 509–512.Google Scholar
[4] Bergeron, F., Flajolet, P. and Salvy, B. (1992) Varieties of increasing trees. In CAAP '92: 17th Colloquium on Trees in Algebra and Programming, Vol. 581 of Lecture Notes in Computer Science, Springer, pp. 24–48.Google Scholar
[5] Bertoin, J. (2006) Random Fragmentation and Coagulation Processes, Cambridge University Press.Google Scholar
[6] Biggins, J. D. (1976) The first- and last-birth problems for a multitype age-dependent branching process. Adv. Appl. Probab. 8 446–459.Google Scholar
[7] Biggins, J. D. (1977) Chernoff's theorem in the branching random walk. J. Appl. Probab. 14 630–636.Google Scholar
[8] Broutin, N. and Devroye, L. (2006) Large deviations for the weighted height of an extended class of trees. Algorithmica 46 271–297.Google Scholar
[9] Broutin, N., Devroye, L., McLeish, E. and de la Salle, M. (2008) The height of increasing trees. Random Struct. Alg. 32 494–518.Google Scholar
[10] Broutin, N. and Holmgren, C. (2012) The total path length of split trees. Ann. Appl. Probab. 22 1745–1777.Google Scholar
[11] Devroye, L. (1999) Universal limit laws for depths in random trees. SIAM J. Comput. 28 409–432.Google Scholar
[12] Drmota, M. (2009) Random Trees, Springer.Google Scholar
[13] Eggenberger, F. and Polya, G. (1923) Über die Statistik verketteter Vorgänge. Zeitschrift Angew. Math. Mech. 3 279–289.Google Scholar
[14] Hoeffding, W. (1961) The strong law of large numbers for U-statistics. Institute of Statistics, University of North Carolina, Mimeograph series 302. https://repository.lib.ncsu.edu/handle/1840.4/2128Google Scholar
[15] Holmgren, C. (2012) Novel characteristic of split trees by use of renewal theory. Electron. J. Probab. 17 # 5.Google Scholar
[16] Holmgren, C. and Janson, S. (2017) Fringe trees, Crump–Mode–Jagers branching processes and m-ary search trees. Probab. Surveys 14 53–154.Google Scholar
[17] Janson, S. (2003) The Wiener index of simply generated random trees. Random Struct. Alg. 22 337–358.Google Scholar
[18] Janson, S. (2004) Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110 177–245.Google Scholar
[19] Jiřina, M. (1958) Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 292–313.Google Scholar
[20] Johnson, N. L. and Kotz, S. (1977) Urn Models and Their Application: An Approach to Modern Discrete Probability Theory, Wiley.Google Scholar
[21] Kallenberg, O. (2002) Foundations of Modern Probability, second edition, Springer.Google Scholar
[22] Kingman, J. F. C. (1978) The representation of partition structures. J. London Math. Soc. (2) 18 374–380.Google Scholar
[23] Kingman, J. F. C. (1982) The coalescent. Stochastic Process. Appl. 13 235–248.Google Scholar
[24] Markov, A. A. (1917) Sur quelques formules limites du calcul des probabilités (Russian). Bulletin de l'Académie Impériale des Sciences, Petrograd 11 177–186.Google Scholar
[25] Neininger, R. (2002) The Wiener index of random trees. Combin. Probab. Comput. 11 587–597.Google Scholar
[26] Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W., eds (2010) NIST Handbook of Mathematical Functions, Cambridge University Press. Also available as NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/Google Scholar
[27] Panholzer, A. and Prodinger, H. (2007) Level of nodes in increasing trees revisited. Random Struct. Alg. 31 203–226.Google Scholar
[28] Pitman, J. (2006) Combinatorial Stochastic Processes, École d'Été de Probabilités de Saint-Flour XXXII, 2002, Vol. 1875 of Lecture Notes in Mathematics, Springer.Google Scholar
[29] Pittel, B. (1994) Note on the heights of random recursive trees and random m-ary search trees. Random Struct. Alg. 5 337–347.Google Scholar
[30] Pólya, G. (1930) Sur quelques points de la théorie des probabilités. Ann. Inst. Poincaré 1 117–161.Google Scholar
[31] Szymański, J. (1987) On a nonuniform random recursive tree. Ann. Discrete Math. 33 297–306.Google Scholar