Abstract
The analysis of an algorithm by Koda and Ruskey for listing ideals in a forest poset leads to a study of random binary trees and their limits as infinite random binary trees. The corresponding finite and infinite random forests are studied too. The infinite random binary trees and forests studied here have exactly one infinite path; they can be defined using suitable size-biazed GaltonWatson processs. Limit theorems are proved using a version of the contraction method.
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References
D. Aldous (1991) The continuum random tree II: an overview. In Proc. Durham Symp. Stoch. Analysis 1990, Cambridge Univ. Press, 23–70.
D. Aldous (1993) The continuum random tree III. Ann. Probab. 21, no. 1, 248–289.
K.B. Athreya & P.E. Ney (1972) Branching Processes. Springer-Verlag, Berlin,1972.
P. Billingsley (1968) Convergence of Probability Measures. Wiley, New York.
T.E. Harris (1951) Some mathematical models for branching processes. In Second Berkeley Symposium on Mathematical Statistics and Probability, Univ. California Press, Berkeley, 305–328.
R.L. Graham, D.E. Knuth & O. Patashnik (1994) Concrete Mathematics. 2nd ed., Addison-Wesley, Reading, Mass.
D.P. Kennedy (1975) The Galton-Watson process conditioned on the total progeny. J. Appl. Probab. 12, 800–806.
M. Klazar (1997) Twelve countings with rooted plane trees. European J. Combin. 18, no. 2, 195–210. Addendum: European J. Combin. 18, no. 6, 739–740.
D.E. Knuth (1997) The Art of Computer Programming. Vol. 1: Fundamental algorithms. 3rd ed., Addison-Wesley, Reading, Mass.
D.E.Knuth, KODA-RUSKEY. CWEB program. Available from http://Sunburn.Stanford.EDU/~knuth/programs/koda-ruskey.w/~knuth/programs/koda-ruskey.w
Y. Koda & F. Ruskey (1993), A Gray code for the ideals of a forest poset. J. Algorithms 15, no. 2, 324–340.
J.-F. Le Gall (1986) Une approche élémentaire des théorèmes de décomposition de Williams. In Séminaire de Probabilités, XX, 1984/85, Lecture Notes in Math., 1204,Springer, Berlin, 447–464.
R. Lyons, R. Pemantle & Y. Peres (1995) Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Probab. 23, no. 3, 1125–1138.
J.-F. Marckert & A. Mokkadem (2001) The depth first processes of Galton-Watson trees converge to the same Brownian excursion. To appear in Ann. Probab.
S.T. Rachev (1991) Probability metrics and the stability of stochastic models. Wiley, Chichester, U.K.
S.T. Rachev & L. Rüschendorf (1995) Probability metrics and recursive algorithms Adv. Appl. Probab. 27, 770–799.
U. Rösler (1991), A limit theorem for “Quicksort”. RAIRO Inform. Théor. Appl. 25, no. 1,85–100.
U. Rösler (1992), A fixed point theorem for distributions. Stochastic Process. Appl. 42, no. 2, 195–214.
U. Rösler & L. Rüschendorf (2001) The contraction method for recursive algorithms. Algorithmica 29, 3–33.
F. Ruskey (1981) Listing and counting subtrees of a tree. SIAM J. Comput. 10, no. 1, 141–150.
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Janson, S. (2002). Ideals in a Forest, One-Way Infinite Binary Trees and the Contraction Method. In: Chauvin, B., Flajolet, P., Gardy, D., Mokkadem, A. (eds) Mathematics and Computer Science II. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8211-8_24
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DOI: https://doi.org/10.1007/978-3-0348-8211-8_24
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9475-3
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