• Bernoulli
  • Volume 23, Number 4A (2017), 2887-2916.

From trees to seeds: On the inference of the seed from large trees in the uniform attachment model

Sébastien Bubeck, Ronen Eldan, Elchanan Mossel, and Miklós Z. Rácz

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study the influence of the seed in random trees grown according to the uniform attachment model, also known as uniform random recursive trees. We show that different seeds lead to different distributions of limiting trees from a total variation point of view. To do this, we construct statistics that measure, in a certain well-defined sense, global “balancedness” properties of such trees. Our paper follows recent results on the same question for the preferential attachment model.

Article information

Bernoulli, Volume 23, Number 4A (2017), 2887-2916.

Received: July 2015
First available in Project Euclid: 9 May 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

random trees seed tree statistical inference uniform attachment


Bubeck, Sébastien; Eldan, Ronen; Mossel, Elchanan; Rácz, Miklós Z. From trees to seeds: On the inference of the seed from large trees in the uniform attachment model. Bernoulli 23 (2017), no. 4A, 2887--2916. doi:10.3150/16-BEJ831.

Export citation


  • [1] Bubeck, S., Mossel, E. and Rácz, M.Z. (2015). On the influence of the seed graph in the preferential attachment model. IEEE Trans. Network Sci. Eng. 2 30–39.
  • [2] Curien, N., Duquesne, T., Kortchemski, I. and Manolescu, I. (2015). Scaling limits and influence of the seed graph in preferential attachment trees. J. Éc. Polytech. Math. 2 1–34.
  • [3] Devroye, L. and Janson, S. (2011). Long and short paths in uniform random recursive dags. Ark. Mat. 49 61–77.
  • [4] Drmota, M. (2009). Random Trees. An interplay Between Combinatorics And Probability. New York: Springer.
  • [5] Evans, S.N., Grübel, R. and Wakolbinger, A. (2012). Trickle-down processes and their boundaries. Electron. J. Probab. 17 1–58.
  • [6] Grübel, R. and Michailow, I. (2015). Random recursive trees: A boundary theory approach. Electron. J. Probab. 20 1–22.
  • [7] Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions. Vol. 2, 2nd ed. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: Wiley.
  • [8] Mahmoud, H.M. (1992). Distances in random plane-oriented recursive trees. J. Comput. Appl. Math. 41 237–245.