Growing Random Uniform d-ary Trees

Let \({{\mathcal {T}}}_{d}(n)\) be the set of d-ary rooted trees with n internal nodes. We give a method to construct a sequence \(( \textbf{t}_{n},n\ge 0)\), where, for any \(n\ge 1\), \( \textbf{t}_{n}\) has the uniform distribution in \({{\mathcal {T}}}_{d}(n)\), and \( \textbf{t}_{n}\) is constructed from \( \textbf{t}_{n-1}\) by the addition of a new node, and a rearrangement of the structure of \( \textbf{t}_{n-1}\). This method is inspired by Rémy’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 \({{\mathcal {T}}}_{d}(n)\) with a cost linear in n.