Degree distribution in the lower levels of the uniform recursive tree

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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