  We introduce a random graph model based on k-trees, which can be generated by
applying a probabilistic preferential attachment rule, but which also has a
simple combinatorial description. We carry out a precise distributional
analysis of important parameters for the network model such as the degree, the
local clustering coefficient and the number of descendants of the nodes and
root-to-node distances. We do not only obtain results for random nodes, but in
particular we also get a precise description of the behaviour of parameters for
the j-th inserted node in a random k-tree of size n, where j = j(n) might grow
with n. The approach presented is not restricted to this specific k-tree model,
but can also be applied to other evolving k-tree models.
