Scaling limits of k-ary growing trees

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 Remy. Our main result concerns the asymptotic behavior of these trees as the number of steps n of the algorithm becomes large: for all k, the sequence of k-ary trees grows at speed n(1/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.