[1] Addario-Berry, L., Devroye, L. and Janson, S. (2013) Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees. Ann. Probab. 41 1072–1087.

[2] Albert, M., Holmgren, C., Johansson, T. and Skerman, F. (2018) Permutations in binary trees and split trees. In *29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)*, Vol. 110 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl–Leibniz-Zentrum für Informatik.

[3] Aldous, D. (1991) The continuum random tree I. Ann. Probab. 19 1–28.

[4] Aldous, D. (1991) The continuum random tree II: An overview. In Stochastic Analysis: Proceedings of the Durham Symposium, 1990, Vol. 167 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 23–70.

[5] Aldous, D. (1993) The continuum random tree III. Ann. Probab. 21 248–289.

[6] Baeza-Yates, R. A. (1987) Some average measures in *m*-ary search trees. Inform. Process. Lett. 25 375–381.

[7] Bender, E. A. (1973) Central and local limit theorems applied to asymptotic enumeration. J. Combin. Theory Ser. A 15 91–111.

[8] Bienaymé, I. J. (1845) De la loi de multiplication et de la durée des familles. *Société Philomatique Paris*, 1845. Reprinted in D. G. Kendall (1975), The genealogy of genealogy branching processes before (and after) 1873, Bull. London Math. Soc. 7 225–253.

[9] Broutin, N. and Holmgren, C. (2012) The total path length of split trees. Ann. Appl. Probab. 22 1745–1777.

[10] Cai, X. S. and Devroye, L. (2017) A study of large fringe and non-fringe subtrees in conditional Galton–Watson trees. Latin American Journal of Probability and Mathematical Statistics XIV 579–611.

[11] Chauvin, B. and Pouyanne, N. (2004) *m*-ary search trees when *m* ≥ 27: A strong asymptotics for the space requirements. Random Struct. Alg. 24 133–154.

[12] Coffman, E. G. Jr., and Eve, J. (1970) File structures using hashing functions. Commun. Assoc. Comput. Mach. 13 427–432.

[14] Cramer, G. (1750) *Introduction à l'Analyse des Lignes Courbes Algébriques*, Frères Cramer et Claude Philibert.

[15] Devroye, L. (1993) On the expected height of fringe-balanced trees. Acta Inform. 30 459–466.

[16] Devroye, L. (1999) Universal limit laws for depths in random trees. SIAM J. Comput. 28 409–432.

[17] Devroye, L., Holmgren, C. and Sulzbach, H. (2017) The heavy path approach to Galton–Watson trees with an application to Apollonian networks. arXiv:1701.02527

[18] DLMF (2017) *NIST Digital Library of Mathematical Functions* (Olver, F. W. J.et al., eds), Release 1.0.15 of 2017-06-01. http://dlmf.nist.gov/ [19] Drmota, M., Iksanov, A., Moehle, M. and Roesler, U. (2009) A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Struct. Alg. 34 319–336.

[20] Durrett, R. (2010) Probability: Theory and Examples, fourth edition, Vol. 31 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.

[21] Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. I, third edition, Wiley.

[22] Finkel, R. A. and Bentley, J. L. (1974) Quad trees a data structure for retrieval on composite keys. Acta Informatica 4 1–9.

[23] Flajolet, P., Poblete, P., and Viola, A. (1998) On the analysis of linear probing hashing. Algorithmica 22 490–515.

[24] Flajolet, P., Roux, M. and Vallée, B. (2010) Digital trees and memoryless sources: From arithmetics to analysis. In *21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10)*, DMTCS Proc. AM, pp. 233–260.

[25] Gessel, I. M., Sagan, B. E. and Yeh, Y. N. (1995) Enumeration of trees by inversions. J. Graph Theory 19 435–459.

[26] Gittenberger, B. (2003) A note on: ‘State spaces of the snake and its tour: Convergence of the discrete snake’. J. Theoret. Probab. 16 1063–1067 (2004), 2003.

[27] Gut, A. (2013) Probability: A Graduate Course, second edition, Springer Texts in Statistics, Springer.

[28] Hoare, C. A. R. (1962) Quicksort. Comput. J. 5 10–15.

[29] Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.

[30] Holmgren, C. (2012) Novel characteristic of split trees by use of renewal theory. Electron. J. Probab. 17 5, 1–27.

[31] Janson, S. (2003) The Wiener index of simply generated random trees. Random Struct. Alg. 22 337–358.

[32] Janson, S. (2012) Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation: Extended abstract. In *23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)*, DMTCS Proc. AQ, pp. 479–490.

[33] Janson, S. (2016) Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees. Random Struct. Alg. 48 57–101.

[34] Janson, S. (2017) Random recursive trees and preferential attachment trees are random split trees. arXiv:1706.05487

[35] Janson, S. and Chassaing, P. (2004) The center of mass of the ISE and the Wiener index of trees. Electron. Comm. Probab. 9 178–187.

[36] Janson, S. and Marckert, J.-F. (2005) Convergence of discrete snakes. J. Theoret. Probab. 18 615–647.

[37] Knuth, D. E. (1998) The Art of Computer Programming, Vol. 3, Addison-Wesley.

[38] Kortchemski, I. (2017) Sub-exponential tail bounds for conditioned stable Bienaymé–Galton–Watson trees. Probab. Theory Rel. Fields 168 1–40.

[39] Louchard, G. and Prodinger, H. (2003) The number of inversions in permutations: A saddle point approach. J. Integer Seq. 6 03.2.8.

[40] Mahmoud, H. M. and Pittel, B. (1989) Analysis of the space of search trees under the random insertion algorithm. J. Algorithms 10 52–75.

[41] Mallows, C. L. and Riordan, J. (1968) The inversion enumerator for labeled trees. Bull. Amer. Math. Soc. 74 92–94.

[42] Margolius, B. H. (2001) Permutations with inversions. J. Integer Seq. 4 01.2.4.

[43] Neininger, R. (2001) On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Struct. Alg. 19 498–524.

[44] Neininger, R. and Rüschendorf, L. (1999) On the internal path length of *d*-dimensional quad trees. Random Struct. Alg. 15 25–41.

[45] Neininger, R. and Rüschendorf, L. (2004) A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 378–418.

[46] Panholzer, A. and Seitz, G. (2012) Limiting distributions for the number of inversions in labelled tree families. Ann. Combin. 16 847–870.

[47] Pyke, R. (1965) Spacings (with discussion). J. Roy. Statist. Soc. Ser. B 27 395–449.

[48] Régnier, M. and Jacquet, P. (1989) New results on the size of tries. IEEE Trans. Inform. Theory 35 203–205.

[49] Rösler, U. (1991) A limit theorem for ‘Quicksort’. RAIRO Inform. Théor. Appl. 25 85–100.

[50] Rösler, U. (2001) On the analysis of stochastic divide and conquer algorithms. Algorithmica 29 238–261.

[51] Rösler, U. and Rüschendorf, L. (2001) The contraction method for recursive algorithms. Algorithmica 29 3–33.

[52] Sachkov, V. N. (1997) Probabilistic Methods in Combinatorial Analysis (translated from the Russian), Vol. 56 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.

[53] Smith, P. J. (1995) A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. Amer. Statist. 49 217–218.

[54] Walker, A. and Wood, D. (1976) Locally balanced binary trees. Comput. J. 19 322–325.

[55] Watson, H. W. and Galton, F. (1875) On the probability of the extinction of families. J. Anthropological Institute of Great Britain and Ireland 4 138–144.