  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.
