Ordering and Self-Similarity in Non-Binary Trees
Mathematics & Statistics
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Ordering and self-similarity in binary trees has been well-studied. Self-similarity was first observed in river networks, and has since been shown to have many useful properties and modeling applications, from leaf vein structures to level-set trees of symmetric Markov chains. Systems for assigning orders to the vertices of non-binary trees, though, have not been the subject of such thorough inquiry, despite potential applications in modeling stochastic processes. The Horton-Strahler ordering for binary trees has been central to the study of self-similarity in random trees. We introduce two classes of multiplier-sum orderings which generalize the Horton-Strahler ordering for binary trees to the space of trees with arbitrary branching and demonstrate that each has an associated pruning process, along with proving some basic properties of these orderings and relationships between them. We also present the results of preliminary numerical simulations which demonstrate the potential validity of Horton law in non-binary trees under these orderings.