Abstract
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.
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Notes
Again \(i{\textbf {f}}^{(i)}\) is the set by adding i as a prefix to all the nodes of \({\textbf {f}}^{(i)}\), so that in \(\textbf{t}_{n+1}\) the subtree rooted at i is isomorphic to \({\textbf {f}}^{(i)}\).
It could be also natural to assume that this cost in \(O(\log n)\), to take into account the size of the pointers.
An extra cost of \(O(\log n)\) to take into account the bit cost of this random generation is also a natural model.
A cost \(O(\log n)\) is also a natural model.
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Communicated by Frédérique Bassino.
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Marckert, JF. Growing Random Uniform d-ary Trees. Ann. Comb. 27, 51–66 (2023). https://doi.org/10.1007/s00026-022-00621-3
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DOI: https://doi.org/10.1007/s00026-022-00621-3