## The Annals of Probability

### Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees

#### Abstract

We study the height and width of a Galton–Watson tree with offspring distribution $\xi$ satisfying $\mathbb{E} \xi=1$, $0<\operatorname{Var} \xi<\infty$, conditioned on having exactly $n$ nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level $k$, for $1\leq k\leq n$.

#### Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 1072-1087.

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.aop/1362750951

Digital Object Identifier
doi:10.1214/12-AOP758

Mathematical Reviews number (MathSciNet)
MR3077536

Zentralblatt MATH identifier
1278.60128

#### Citation

Addario-Berry, Louigi; Devroye, Luc; Janson, Svante. Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees. Ann. Probab. 41 (2013), no. 2, 1072--1087. doi:10.1214/12-AOP758. https://projecteuclid.org/euclid.aop/1362750951

#### References

• [1] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Mathematical Society Lecture Note Series 167 23–70. Cambridge Univ. Press, Cambridge.
• [2] Aldous, D. and Pitman, J. (1998). Tree-valued Markov chains derived from Galton–Watson processes. Ann. Inst. Henri Poincaré Probab. Stat. 34 637–686.
• [3] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
• [4] Biane, P., Pitman, J. and Yor, M. (2001). Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. (N.S.) 38 435–465 (electronic).
• [5] Chassaing, P., Marckert, J. F. and Yor, M. (2000). The height and width of simple trees. In Mathematics and Computer Science (Versailles, 2000) 17–30. Birkhäuser, Basel.
• [6] Chung, K. L. (1975). Maxima in Brownian excursions. Bull. Amer. Math. Soc. 81 742–745.
• [7] de Bruijn, N. G., Knuth, D. E. and Rice, S. O. (1972). The average height of planted plane trees. In Graph Theory and Computing 15–22. Academic Press, New York.
• [8] Devroye, L. (1998). Branching processes and their applications in the analysis of tree structures and tree algorithms. In Probabilistic Methods for Algorithmic Discrete Mathematics, (Habib, McDiarmid, Ramirez andReed, eds.). Algorithms Combin. 16 249–314. Springer, Berlin.
• [9] Donati-Martin, C. (2001). Some remarks about the identity in law for the Bessel bridge $\int^{1}_{0}\frac{ds}{r(s)}\stackrel{({\rm law})}{\to}=2\sup_{s\leq1}r(s)$. Studia Sci. Math. Hungar. 37 131–144.
• [10] Drmota, M. (2009). Random Trees: An Interplay Between Combinatorics and Probability. Springer, New York.
• [11] Drmota, M. and Gittenberger, B. (1997). On the profile of random trees. Random Structures Algorithms 10 421–451.
• [12] Drmota, M. and Gittenberger, B. (2004). The width of Galton–Watson trees conditioned by the size. Discrete Math. Theor. Comput. Sci. 6 387–400 (electronic).
• [13] Duquesne, T. (2003). A limit theorem for the contour process of conditioned Galton–Watson trees. Ann. Probab. 31 996–1027.
• [14] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi+147.
• [15] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553–603.
• [16] Dwass, M. (1969). The total progeny in a branching process and a related random walk. J. Appl. Probab. 6 682–686.
• [17] Flajolet, P., Gao, Z., Odlyzko, A. and Richmond, B. (1993). The distribution of heights of binary trees and other simple trees. Combin. Probab. Comput. 2 145–156.
• [18] Flajolet, P. and Odlyzko, A. (1982). The average height of binary trees and other simple trees. J. Comput. System Sci. 25 171–213.
• [19] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
• [20] Janson, S. (2006). Random cutting and records in deterministic and random trees. Random Structures Algorithms 29 139–179.
• [21] Janson, S. (2006). Conditioned Galton–Watson trees do not grow. In Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities 331–334. Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
• [22] Janson, S. (2008). On the asymptotic joint distribution of height and width in random trees. Studia Sci. Math. Hungar. 45 451–467.
• [23] Janson, S. (2012). Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probab. Surv. 9 103–252.
• [24] Kennedy, D. P. (1975). The Galton–Watson process conditioned on the total progeny. J. Appl. Probab. 12 800–806.
• [25] Kennedy, D. P. (1976). The distribution of the maximum Brownian excursion. J. Appl. Probab. 13 371–376.
• [26] Kersting, G. (1998). On the height profile of a conditioned Galton–Watson tree. Preprint. Available at http://ismi.math.uni-frankfurt.de/kersting/research/profile.ps.
• [27] Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré Probab. Stat. 22 425–487.
• [28] Kolchin, V. F. (1986). Random Mappings. Optimization Software Inc. Publications Division, New York.
• [29] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311.
• [30] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
• [31] McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics, (Habib, McDiarmid, Ramirez andReed, eds.). Algorithms Combin. 16 195–248. Springer, Berlin.
• [32] Meir, A. and Moon, J. W. (1978). On the altitude of nodes in random trees. Canad. J. Math. 30 997–1015.
• [33] Pitman, J. (1998). Enumerations of trees and forests related to branching processes and random walks. In Microsurveys in Discrete Probability (Princeton, NJ, 1997). DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 41 163–180. Amer. Math. Soc., Providence, RI.
• [34] Rényi, A. and Szekeres, G. (1967). On the height of trees. J. Aust. Math. Soc. 7 497–507.