The height of scaled attachment random recursive trees

Abstract : We study depth properties of a general class of random recursive trees where each node $n$ attaches to the random node $\lfloor nX_n \rfloor$ and $X_0, \ldots , X_n$ is a sequence of i.i.d. random variables taking values in $[0,1)$. We call such trees scaled attachment random recursive trees (SARRT). We prove that the height $H_n$ of a SARRT is asymptotically given by $H_n \sim \alpha_{\max} \log n$ where $\alpha_{\max}$ is a constant depending only on the distribution of $X_0$ whenever $X_0$ has a bounded density. This gives a new elementary proof for the height of uniform random recursive trees $H_n \sim e \log n$ that does not use branching random walks.
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Drmota, Michael and Gittenberger, Bernhard. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), pp.129-142, 2010, DMTCS Proceedings
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Luc Devroye, Omar Fawzi, Nicolas Fraiman. The height of scaled attachment random recursive trees. Drmota, Michael and Gittenberger, Bernhard. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), pp.129-142, 2010, DMTCS Proceedings. 〈hal-01185586〉

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