A non-increasing tree growth process for recursive trees and applications

We introduce a non-increasing tree growth process $((T_n,\sigma_n),\, n\ge 1)$, where $T_n$ is a rooted labeled tree on $n$ vertices and ${\sigma}_n$ is a permutation of the vertex labels. The construction of $(T_{n},{\sigma}_n)$ from $(T_{n-1},{\sigma}_{n-1})$ involves rewiring a random (possibly empty) subset of edges in $T_{n-1}$ towards the newly added vertex; as a consequence $T_{n-1} \not\subset T_n$ with positive probability. The key feature of the process is that the shape of $T_n$ has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotonous in the process. We present two applications. First, while couplings between Kingman's coalescent and random recursive trees where known for any fixed $n$, this new process provides a non-standard coupling of all finite Kingman's coalescents. Second, we use the new process and the Chen-Stein method to extend the well-understood properties of degree distribution of random recursive trees to extremal-range cases. Namely, we obtain convergence rates on the number of vertices with degree at least $c\ln n$, $c\in (1,2)$, in trees with $n$ vertices. Further avenues of research are discussed.