For each integer k≥2, we introduce a sequence of k-ary discrete
trees constructed recursively by choosing at each step an edge uniformly among
the present edges and grafting on "its middle" k−1 new edges. When k=2,
this corresponds to a well-known algorithm which was first introduced by
R\'emy. Our main result concerns the asymptotic behavior of these trees as n
becomes large: for all k, the sequence of k-ary trees grows at speed
n1/k towards a k-ary random real tree that belongs to the family of
self-similar fragmentation trees. This convergence is proved with respect to
the Gromov-Hausdorff-Prokhorov topology. We also study embeddings of the
limiting trees when k varies
