Degree distribution in the lower levels of the uniform recursive tree

Details Degree distribution in the lower levels of the uniform recursive tree Degree distribution in the lower levels of the uniform recursive tree

2011-12-06

arxiv.org/abs/1112.1250v1

Mathematics

Probability

8 pages

Scientific paper

In this note we consider the $k$th level of the uniform random recursive tree after $n$ steps, and prove that the proportion of nodes with degree greater than $t\log n$ converges to $(1-t)^k$ almost surely, as $n\to\infty$, for every $t\in(0,1)$. In addition, we show that the number of degree $d$ nodes in the first level is asymptotically Poisson distributed with mean 1; moreover, they are asymptotically independent for $d=1,2,...$.

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