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The distribution of nodes of given degree in random trees

✍ Scribed by Drmota, Michael; Gittenberger, Bernhard

Publisher: John Wiley and Sons
Year: 1999
Tongue: English
Weight: 391 KB
Volume: 31
Category: Article
ISSN: 0364-9024
DOI: 10.1002/(sici)1097-0118(199907)31:3<227::aid-jgt6>3.0.co;2-6

No coin nor oath required. For personal study only.

✦ Synopsis

Let T n denote the set of unrooted unlabeled trees of size n and let k ≥ 1 be given. By assuming that every tree of T n is equally likely, it is shown that the limiting distribution of the number of nodes of degree k is normal with mean value ∼ µ k n and variance ∼ σ 2 k n with positive constants µ k and σ k . Besides, the asymptotic behavior of µ k and σ k for k → ∞ as well as the corresponding multivariate distributions are derived. Furthermore, similar results can be proved for plane trees, for labeled trees, and for forests.

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