Limit distribution for the maximum degree of a random recursive tree

Authors:
William Goh;Eric Schmutz
Affiliations:
MCS Department, Drexel University, Philadelphia, PA;MCS Department, Drexel University, Philadelphia, PA
Venue:
Journal of Computational and Applied Mathematics - Special issue: Probabilistic methods in combinatorics and combinatorial optimization
Year:
2002

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Abstract

If a recursive tree is selected uniformly at random from among all recursive trees on n vertices, then the distribution of the maximum in-degree Δ is given asymptotically by the following theorem: for any fixed integer d, Pn(Δ ≤ [µn] + d) = exp(-2{µn}-d-1) + o(1) as n → ∞ , where µn = log2n. (As usual, [µn] denotes the greatest integer less than or equal to µn, and {µn} = µn- [µn].) The proof makes extensive use of asymptotic approximations for the partial sums of the exponential series.