Growing Random Uniform d-ary Trees

Let T-d(n) be the set of d-ary rooted trees with n internal nodes. We give a method to construct a sequence (t(n), n >= 0), where, for any n >= 1, t(n) has the uniform distribution in T-d(n), and t(n) is constructed from t(n)(-1) by the addition of a new node, and a rearrangement of the structure of t(n-1). This method is inspired by Remy's algorithm which does this job in the binary case, but it is different from it. This provides a method for the random generation of a uniform d-ary tree in T-d(n) with a cost linear in n.