{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T10:18:07Z","timestamp":1785406687258,"version":"3.56.0"},"reference-count":73,"publisher":"American Physical Society (APS)","issue":"6","license":[{"start":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T00:00:00Z","timestamp":1764547200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/link.aps.org\/licenses\/aps-default-license"}],"funder":[{"DOI":"10.13039\/100031478","name":"NextGenerationEU","doi-asserted-by":"publisher","award":["PNRR_M4C2I4.1._DM351"],"award-info":[{"award-number":["PNRR_M4C2I4.1._DM351"]}],"id":[{"id":"10.13039\/100031478","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100031478","name":"NextGenerationEU","doi-asserted-by":"publisher","award":["CN_00000013_CN-HPC"],"award-info":[{"award-number":["CN_00000013_CN-HPC"]}],"id":[{"id":"10.13039\/100031478","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100031478","name":"NextGenerationEU","doi-asserted-by":"publisher","award":["M4C2I1.4"],"award-info":[{"award-number":["M4C2I1.4"]}],"id":[{"id":"10.13039\/100031478","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. E"],"accepted":{"date-parts":[[2025,11,10]]},"DOI":"10.1103\/qmv9-1czv","type":"journal-article","created":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T00:36:26Z","timestamp":1762821386000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Configurational entropy of random trees"],"prefix":"10.1103","volume":"112","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-1527-3840","authenticated-orcid":true,"given":"Pieter H. W.","family":"van der Hoek","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/004fze387","id-type":"ROR","asserted-by":"publisher"}],"name":"Scuola Internazionale Superiore di Studi Avanzati"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9627-2486","authenticated-orcid":true,"given":"Angelo","family":"Rosa","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/004fze387","id-type":"ROR","asserted-by":"publisher"}],"name":"Scuola Internazionale Superiore di Studi Avanzati"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6843-2753","authenticated-orcid":true,"given":"Ralf","family":"Everaers","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00w5ay796","id-type":"ROR","asserted-by":"publisher"}],"name":"ENS de Lyon"},{"id":[{"id":"https:\/\/ror.org\/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https:\/\/ror.org\/04zmssz18","id-type":"ROR","asserted-by":"publisher"}],"name":"LPENSL UMR5672"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2025,12,1]]},"reference":[{"key":"qmv9-1czvCc1R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.40.1610","type":"journal-article"},{"key":"qmv9-1czvCc2R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.20.2785","type":"journal-article"},{"key":"qmv9-1czvCc3R1","doi-asserted-by":"publisher","DOI":"10.1051\/jphyslet:019800041019046900","type":"journal-article"},{"key":"qmv9-1czvCc4R1","doi-asserted-by":"publisher","DOI":"10.1051\/jphys:0198100420120158500","type":"journal-article"},{"key":"qmv9-1czvCc5R1","volume-title":"Critical Phenomena, Random Systems, Gauge Theories","author":"A. Bovier","year":"1984","type":"book"},{"key":"qmv9-1czvCc6R1","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-49780-3_3","type":"journal-article"},{"key":"qmv9-1czvCc7R1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198520597.001.0001","type":"book","volume-title":"Polymer physics","author":"M. Rubinstein","year":"2003"},{"key":"qmv9-1czvCc8R1","doi-asserted-by":"publisher","DOI":"10.1021\/ja01856a062","type":"journal-article"},{"key":"qmv9-1czvCc9R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1747157","type":"journal-article"},{"key":"qmv9-1czvCc10R1","doi-asserted-by":"publisher","DOI":"10.1002\/bip.1968.360060508","type":"journal-article"},{"key":"qmv9-1czvCc11R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.20.2130","type":"journal-article"},{"key":"qmv9-1czvCc12R1","doi-asserted-by":"publisher","DOI":"10.1051\/jphys:01981004202017500","type":"journal-article"},{"key":"qmv9-1czvCc13R1","doi-asserted-by":"publisher","DOI":"10.1051\/jphys:0198100420100135900","type":"journal-article"},{"key":"qmv9-1czvCc14R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.46.871","type":"journal-article"},{"key":"qmv9-1czvCc15R1","doi-asserted-by":"publisher","DOI":"10.1016\/0378-4371(92)90075-2","type":"journal-article"},{"key":"qmv9-1czvCc16R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.464279","type":"journal-article"},{"key":"qmv9-1czvCc17R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.74.809","type":"journal-article"},{"key":"qmv9-1czvCc18R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.51.2213","type":"journal-article"},{"key":"qmv9-1czvCc19R1","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/48\/34\/345003","type":"journal-article"},{"key":"qmv9-1czvCc20R1","doi-asserted-by":"publisher","DOI":"10.1063\/5.0287899","type":"journal-article"},{"key":"qmv9-1czvCc21R1","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/12\/9\/004","type":"journal-article"},{"key":"qmv9-1czvCc22R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.441870","type":"journal-article"},{"key":"qmv9-1czvCc23R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01020605","type":"journal-article"},{"key":"qmv9-1czvCc24R1","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/25\/2\/013","type":"journal-article"},{"key":"qmv9-1czvCc25R1","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/38\/4\/001","type":"journal-article"},{"key":"qmv9-1czvCc26R1","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/49\/34\/345001","type":"journal-article"},{"key":"qmv9-1czvCc27R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.4965827","type":"journal-article"},{"key":"qmv9-1czvCc28R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.95.012117","type":"journal-article"},{"key":"qmv9-1czvCc29R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.104.014501","type":"journal-article"},{"key":"qmv9-1czvCc30R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.110.045312","type":"journal-article"},{"key":"qmv9-1czvCc31R1","doi-asserted-by":"publisher","DOI":"10.1021\/ma00058a016","type":"journal-article"},{"key":"qmv9-1czvCc32R1","doi-asserted-by":"publisher","DOI":"10.1039\/C6SM02756C","type":"journal-article"},{"key":"qmv9-1czvCc33R1","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(85)90678-4","type":"journal-article"},{"key":"qmv9-1czvCc34R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.57.3023","type":"journal-article"},{"key":"qmv9-1czvCc35R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.73.1263","type":"journal-article"},{"key":"qmv9-1czvCc36R1","doi-asserted-by":"publisher","DOI":"10.1039\/C3SM52805G","type":"journal-article"},{"key":"qmv9-1czvCc37R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.112.118302","type":"journal-article"},{"key":"qmv9-1czvCc38R1","doi-asserted-by":"publisher","DOI":"10.1039\/C6SM02168A","type":"journal-article"},{"key":"qmv9-1czvCc39R1","doi-asserted-by":"publisher","DOI":"10.1140\/epje\/i2019-11765-3","type":"journal-article"},{"key":"qmv9-1czvCc40R1","doi-asserted-by":"publisher","DOI":"10.1039\/C8SM02587H","type":"journal-article"},{"key":"qmv9-1czvCc41R1","doi-asserted-by":"publisher","DOI":"10.1021\/acsmacrolett.8b00828","type":"journal-article"},{"key":"qmv9-1czvCc42R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.8036491","type":"journal-article"},{"key":"qmv9-1czvCc43R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.52.2912","type":"journal-article"},{"key":"qmv9-1czvCc44R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0301-4622(98)00115-X","type":"journal-article"},{"key":"qmv9-1czvCc45R1","doi-asserted-by":"publisher","DOI":"10.1006\/jsbi.2001.4420","type":"journal-article"},{"key":"qmv9-1czvCc46R1","doi-asserted-by":"publisher","DOI":"10.3389\/fmicb.2023.1192831","type":"journal-article"},{"key":"qmv9-1czvCc47R1","doi-asserted-by":"publisher","DOI":"10.1209\/0295-5075\/23\/5\/012","type":"journal-article"},{"key":"qmv9-1czvCc48R1","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1000153","type":"journal-article"},{"key":"qmv9-1czvCc49R1","doi-asserted-by":"publisher","DOI":"10.1088\/1478-3975\/6\/2\/025013","type":"journal-article"},{"key":"qmv9-1czvCc50R1","doi-asserted-by":"publisher","DOI":"10.1088\/0034-4885\/77\/2\/022601","type":"journal-article"},{"key":"qmv9-1czvCc51R1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0808089105","type":"journal-article"},{"key":"qmv9-1czvCc52R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.3652763","type":"journal-article"},{"key":"qmv9-1czvCc53R1","doi-asserted-by":"publisher","DOI":"10.1063\/5.0152604","type":"journal-article"},{"key":"qmv9-1czvCc54R1","first-page":"376","type":"journal-article","volume":"23","author":"A. Cayley","year":"1889","journal-title":"Q. J. Math."},{"key":"qmv9-1czvCc55R1","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jpcb.6b02258","type":"journal-article"},{"key":"qmv9-1czvCc56R1","first-page":"742","type":"journal-article","volume":"27","author":"H. Pr\u00fcfer","year":"1918","journal-title":"Arch. Math. Phys."},{"key":"qmv9-1czvCc57R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/qmv9-1czv for two tables summarizing the processes of coding and decoding of Pr\u00fcfer codes, derivation of limiting behaviors of branch nodes, generalized version of Kramers&#39; theorem, summary of the de Gennes\u2013Daoud\u2013Joanny theory, one additional figure, and a table.","type":"other"},{"key":"qmv9-1czvCc58R1","doi-asserted-by":"publisher","DOI":"10.1112\/S0025579300004290","type":"journal-article"},{"key":"qmv9-1czvCc59R1","unstructured":"As a check, one recovers Eq.   (2) by summing over all possible combinations of node functionalities,","type":"other"},{"key":"qmv9-1czvCc59R2","unstructured":"\u03a9N=\u2211f1+f2+\u22ef+fN=2(N\u22121)f1,f2,...,fN\u22651\u03a9N(fi),by setting ki=fi\u22121 in the multinomial theorem (see https:\/\/en.wikipedia.org\/wiki\/Multinomial_theorem) which states that for any integer N\u22652,\n  NN\u22122=\u2211k1+k2+\u22ef+kN=N\u22122k1,k2,...,kN\u22650(N\u22122)!k1!k2!\u22efkN!.","type":"other"},{"key":"qmv9-1czvCc60R1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1966-11442-8","type":"journal-article"},{"key":"qmv9-1czvCc61R1","volume-title":"Understanding Molecular Simulation: From Algorithms to Applications","author":"D. Frenkel","year":"2002","type":"book"},{"key":"qmv9-1czvCc62R1","doi-asserted-by":"publisher","DOI":"10.4236\/jsea.2009.22016","type":"journal-article"},{"key":"qmv9-1czvCc63R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1673106","type":"journal-article"},{"key":"qmv9-1czvCc64R1","unstructured":"https:\/\/oeis.org\/A000055\/b000055.txt.","type":"other"},{"key":"qmv9-1czvCc65R1","doi-asserted-by":"publisher","DOI":"10.2307\/1969046","type":"journal-article"},{"key":"qmv9-1czvCc66R1","volume-title":"Combinatorial Algorithms for Computers and Calculators","author":"A. Nijenhuis","year":"1978","type":"book"},{"key":"qmv9-1czvCc67R1","unstructured":"P.\u00a0H.\u00a0W. van der Hoek, A. Rosa, E. Ghobadpour, and R. Everaers (unpublished).","type":"other"},{"key":"qmv9-1czvCc68R1","doi-asserted-by":"publisher","DOI":"10.1021\/acs.macromol.5c00697","type":"journal-article"},{"key":"qmv9-1czvCc69R1","unstructured":"E. Ghobadpour, Multiple scale modeling of bacterial chromosome, Ph.D. thesis, Universit\u00e9 Grenoble Alpes (France), 2023.","type":"other"},{"key":"qmv9-1czvCc70R1","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2006.880730","type":"journal-article"},{"key":"qmv9-1czvCc71R1","unstructured":"Decoding bare so-called Blob or Dandelion codes is also done in linear computing time, but twice as fast compared to decoding bare Pr\u00fcfer codes   [70]. This is in part because a tree can only be reconstructed from a Pr\u00fcfer code, once the functionality fi of each node i is known. When the Pr\u00fcfer code is the only information available, those functionalities can only be obtained by first reading the entire Pr\u00fcfer code itself, which is an O(n) operation. For Blob or Dandelion codes this step is not necessary, because information about the tree connectivity in these codes is stored locally. A first-step attempt in improving our sampling algorithm even further might thus be by adopting Blob or Dandelion codes rather than Pr\u00fcfer codes. However, it must be emphasized that for any sampling algorithm analogous to the one presented here, the functionalities of individual nodes need to be sampled first before a code (Dandelion, Blob, or Pr\u00fcfer) can be sampled. In our setup, we therefore save a step O(n) of computational complexity in decoding Pr\u00fcfer codes compared to the notion used in Ref.   [70].","type":"other"},{"key":"qmv9-1czvCc72R1","author":"P. H. W. van der Hoek","year":"2025","journal-title":"github","type":"dataset"}],"container-title":["Physical Review E"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/qmv9-1czv","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/qmv9-1czv\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T14:03:31Z","timestamp":1764597811000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/qmv9-1czv"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,1]]},"references-count":73,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"URL":"https:\/\/doi.org\/10.1103\/qmv9-1czv","relation":{},"ISSN":["2470-0045","2470-0053"],"issn-type":[{"value":"2470-0045","type":"print"},{"value":"2470-0053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,1]]},"assertion":[{"value":"2025-04-17","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"\u00a92025 American Physical Society. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2025","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"American Physical Society","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"065405"}}