Asymptotic properties of weighted recursive and preferential
attachment trees
Starting from a sequence of positive real numbers $(w_n)$, which
we call weights, we construct a tree in a recursive manner: at time 1,
the tree has only one vertex. Then at any step n+1, we add a new vertex
to the tree and we choose its parent at random among the already
existing vertices, in such a way that the k-th vertex (in order of
creation) is chosen with probability proportional to $w_k$.
This model generalises the well-known uniform recursive tree (URT) in
the case of a constant sequence $(w_n)$. In fact, it can also be shown
that the trees constructed using affine preferential attachment can be
described with this construction, using a random sequence of weights $(w_n)$.
We prove almost-sure scaling limits for the height, profile and degrees
in the tree as the number of vertices tends to infinity. These results
are related to proving scaling limits in the Gromov-Hausdorff-Prokhorov
topology for a family of random growth models on graphs that generalises
Rémy's algorithm.