A relaxed k-ary tree is an ordered directed acyclic graph with a unique source and sink in which every node has out-degree k. These objects arise in the compression of trees in which some repeated subtrees are factored and repeated appearances are replaced by pointers. We prove an asymptotic theta-result for the number of relaxed k-ary tree with n nodes for n → ∞. This generalizes the previously proved binary case to arbitrary finite arity, and shows that the seldom observed phenomenon of a stretched exponential term e^{c n^{1/3}} appears in all these cases. We also derive the recurrences for compacted k-ary trees in which all subtrees are unique and minimal deterministic finite automata accepting a finite language over a finite alphabet.
@InProceedings{ghoshdastidar_et_al:LIPIcs.AofA.2024.15, author = {Ghosh Dastidar, Manosij and Wallner, Michael}, title = {{Asymptotics of Relaxed k-Ary Trees}}, booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)}, pages = {15:1--15:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-329-4}, ISSN = {1868-8969}, year = {2024}, volume = {302}, editor = {Mailler, C\'{e}cile and Wild, Sebastian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.15}, URN = {urn:nbn:de:0030-drops-204506}, doi = {10.4230/LIPIcs.AofA.2024.15}, annote = {Keywords: Asymptotic enumeration, stretched exponential, Airy function, directed acyclic graph, Dyck paths, compacted trees, minimal automata} }
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