{"id":"https://openalex.org/W1996039969","doi":"https://doi.org/10.1017/s0963548300000560","title":"The Distribution of Heights of Binary Trees and Other Simple Trees","display_name":"The Distribution of Heights of Binary Trees and Other Simple Trees","publication_year":1993,"publication_date":"1993-06-01","ids":{"openalex":"https://openalex.org/W1996039969","doi":"https://doi.org/10.1017/s0963548300000560","mag":"1996039969"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548300000560","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548300000560","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://inria.hal.science/inria-00076989","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5018743761","display_name":"Philippe Flajolet","orcid":null},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en informatique et en automatique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Philippe Flajolet","raw_affiliation_strings":["INRIA, 78150 Rocquencourt, France","INRIA, 78150 Rocquencourt, France#TAB#"],"affiliations":[{"raw_affiliation_string":"INRIA, 78150 Rocquencourt, France","institution_ids":["https://openalex.org/I1326498283"]},{"raw_affiliation_string":"INRIA, 78150 Rocquencourt, France#TAB#","institution_ids":["https://openalex.org/I1326498283"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102714122","display_name":"Zhicheng Gao","orcid":"https://orcid.org/0000-0001-6488-8721"},"institutions":[{"id":"https://openalex.org/I67031392","display_name":"Carleton University","ror":"https://ror.org/02qtvee93","country_code":"CA","type":"education","lineage":["https://openalex.org/I67031392"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Zhicheng Gao","raw_affiliation_strings":["Dept. of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S5B6, Canada","(Carleton University)"],"affiliations":[{"raw_affiliation_string":"Dept. of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S5B6, Canada","institution_ids":["https://openalex.org/I67031392"]},{"raw_affiliation_string":"(Carleton University)","institution_ids":["https://openalex.org/I67031392"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068034458","display_name":"Andrew Odlyzko","orcid":null},"institutions":[{"id":"https://openalex.org/I1283103587","display_name":"AT&T (United States)","ror":"https://ror.org/02bbd5539","country_code":"US","type":"company","lineage":["https://openalex.org/I1283103587"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Andrew Odlyzko","raw_affiliation_strings":["AT& T Bell Laboratories, Murray Hill, New Jersey 07974, United States"],"affiliations":[{"raw_affiliation_string":"AT& T Bell Laboratories, Murray Hill, New Jersey 07974, United States","institution_ids":["https://openalex.org/I1283103587"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5051530602","display_name":"Bruce M. Richmond","orcid":"https://orcid.org/0000-0002-0056-5832"},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Bruce Richmond","raw_affiliation_strings":["Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L3G1, Canada","(University of Waterloo)"],"affiliations":[{"raw_affiliation_string":"Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L3G1, Canada","institution_ids":["https://openalex.org/I151746483"]},{"raw_affiliation_string":"(University of Waterloo)","institution_ids":["https://openalex.org/I151746483"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5018743761"],"corresponding_institution_ids":["https://openalex.org/I1326498283"],"apc_list":null,"apc_paid":null,"fwci":0.577,"has_fulltext":false,"cited_by_count":53,"citation_normalized_percentile":{"value":0.59034864,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"2","issue":"2","first_page":"145","last_page":"156"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.8001164197921753},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7983457446098328},{"id":"https://openalex.org/keywords/binary-tree","display_name":"Binary tree","score":0.7273316979408264},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6772431135177612},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.6389939188957214},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.5473886728286743},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.544873058795929},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.5245229005813599},{"id":"https://openalex.org/keywords/binary-search-tree","display_name":"Binary search tree","score":0.442732036113739},{"id":"https://openalex.org/keywords/weight-balanced-tree","display_name":"Weight-balanced tree","score":0.4416918158531189},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.22558501362800598},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.14829307794570923},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.05193394422531128},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.050040870904922485}],"concepts":[{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.8001164197921753},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7983457446098328},{"id":"https://openalex.org/C197855036","wikidata":"https://www.wikidata.org/wiki/Q380172","display_name":"Binary tree","level":2,"score":0.7273316979408264},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6772431135177612},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.6389939188957214},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5473886728286743},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.544873058795929},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.5245229005813599},{"id":"https://openalex.org/C91154448","wikidata":"https://www.wikidata.org/wiki/Q623818","display_name":"Binary search tree","level":3,"score":0.442732036113739},{"id":"https://openalex.org/C57900377","wikidata":"https://www.wikidata.org/wiki/Q7979855","display_name":"Weight-balanced tree","level":4,"score":0.4416918158531189},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.22558501362800598},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.14829307794570923},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.05193394422531128},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.050040870904922485},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1017/s0963548300000560","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548300000560","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},{"id":"pmh:hdl:10068/42803","is_oa":false,"landing_page_url":"http://hdl.handle.net/10068/42803","pdf_url":null,"source":{"id":"https://openalex.org/S4377196900","display_name":"OpenGrey (Institut de l'Information Scientifique et Technique)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1295345811","host_organization_name":"Institut de l'Information Scientifique et Technique","host_organization_lineage":["https://openalex.org/I1295345811"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"R - Report"},{"id":"pmh:oai:HAL:inria-00076989v1","is_oa":true,"landing_page_url":"https://inria.hal.science/inria-00076989","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"[Research Report] RR-1749, INRIA. 1992","raw_type":"Reports"}],"best_oa_location":{"id":"pmh:oai:HAL:inria-00076989v1","is_oa":true,"landing_page_url":"https://inria.hal.science/inria-00076989","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"[Research Report] RR-1749, INRIA. 1992","raw_type":"Reports"},"sustainable_development_goals":[{"score":0.4300000071525574,"id":"https://metadata.un.org/sdg/15","display_name":"Life in Land"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W594514321","https://openalex.org/W1983188796","https://openalex.org/W2004276429","https://openalex.org/W2026252457","https://openalex.org/W2063643785","https://openalex.org/W2066828720","https://openalex.org/W2078333438","https://openalex.org/W2098828803","https://openalex.org/W2123024824","https://openalex.org/W2159376684","https://openalex.org/W2167266655","https://openalex.org/W2168559467","https://openalex.org/W4239614637"],"related_works":["https://openalex.org/W2056975861","https://openalex.org/W1982136501","https://openalex.org/W2013246077","https://openalex.org/W2402045266","https://openalex.org/W2571039174","https://openalex.org/W2025994050","https://openalex.org/W1548477435","https://openalex.org/W4281555315","https://openalex.org/W2060664390","https://openalex.org/W2053972362"],"abstract_inverted_index":{"The":[0,70],"number,":[1],",":[2,35,53],"of":[3,8,77],"rooted":[4],"plane":[5],"binary":[6],"trees":[7],"height":[9],"\u2264":[10,28,30],"h":[11,50],"with":[12],"n":[13,25,32],"internal":[14],"nodes":[15],"is":[16,39,47],"shown":[17],"to":[18,73],"satisfy":[19],"uniformly":[20],"for":[21,46,49,61,65],"\u03b4":[22,38],"\u22121":[23],"(log":[24],")":[26,33],"\u22121/2":[27],"\u03b2":[29],"\u03b4(log":[31],"1/2":[34],"where":[36,54],"and":[37,67,79],"a":[40],"positive":[41],"constant.":[42],"An":[43],"asymptotic":[44,82],"formula":[45],"derived":[48,64],"=":[51],"cn":[52],"0":[55],"&lt;":[56,58],"c":[57],"1.":[59],"Bounds":[60],"are":[62,84],"also":[63],"large":[66],"small":[68],"heights.":[69],"methods":[71],"apply":[72],"any":[74],"simple":[75],"family":[76],"trees,":[78],"the":[80],"general":[81],"results":[83],"stated.":[85]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
