{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T23:54:48Z","timestamp":1767916488819,"version":"3.49.0"},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2008,9,12]],"date-time":"2008-09-12T00:00:00Z","timestamp":1221177600000},"content-version":"unspecified","delay-in-days":5582,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[1993,6]]},"abstract":"<jats:p>The number, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548300000560inline1\"\/>, of rooted plane binary trees of height \u2264 <jats:italic>h<\/jats:italic> with <jats:italic>n<\/jats:italic> internal nodes is shown to satisfy<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548300000560eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>uniformly for \u03b4<jats:sup>\u22121<\/jats:sup>(log <jats:italic>n<\/jats:italic>)<jats:sup>\u22121\/2<\/jats:sup> \u2264 \u03b2 \u2264 \u03b4(log <jats:italic>n<\/jats:italic>)<jats:sup>1\/2<\/jats:sup>, where <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548300000560inline2\"\/> and \u03b4 is a positive constant. An asymptotic formula for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548300000560inline3\"\/> is derived for <jats:italic>h<\/jats:italic> = <jats:italic>cn<\/jats:italic>, where 0 &lt; c &lt; 1. Bounds for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548300000560inline4\"\/> are also derived for large and small heights. The methods apply to any simple family of trees, and the general asymptotic results are stated.<\/jats:p>","DOI":"10.1017\/s0963548300000560","type":"journal-article","created":{"date-parts":[[2008,9,12]],"date-time":"2008-09-12T11:19:08Z","timestamp":1221218348000},"page":"145-156","source":"Crossref","is-referenced-by-count":27,"title":["The Distribution of Heights of Binary Trees and Other Simple Trees"],"prefix":"10.1017","volume":"2","author":[{"given":"Philippe","family":"Flajolet","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhicheng","family":"Gao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrew","family":"Odlyzko","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bruce","family":"Richmond","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2008,9,12]]},"reference":[{"key":"S0963548300000560_ref013","doi-asserted-by":"publisher","DOI":"10.1137\/0215039"},{"key":"S0963548300000560_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/B978-1-4832-3187-7.50007-6"},{"key":"S0963548300000560_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(91)90197-R"},{"key":"S0963548300000560_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0000(82)90004-6"},{"key":"S0963548300000560_ref007","unstructured":"[7] Luczak T. (preprint) The number of trees with a large diameter."},{"key":"S0963548300000560_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(82)90005-6"},{"key":"S0963548300000560_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF02559539"},{"key":"S0963548300000560_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(83)90062-6"},{"key":"S0963548300000560_ref003","volume-title":"Asymptotic Methods in Analysis","author":"de Bruijn","year":"1961"},{"key":"S0963548300000560_ref010","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700004432"},{"key":"S0963548300000560_ref006","unstructured":"[6] Kolchin V. F. (1986) Random Mappings, (English translation of 1984 Russian original), Optimization Software."},{"key":"S0963548300000560_ref002","doi-asserted-by":"publisher","DOI":"10.1287\/moor.9.1.43"},{"key":"S0963548300000560_ref008","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1978-085-0"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548300000560","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T22:49:06Z","timestamp":1557960546000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548300000560\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,6]]},"references-count":13,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1993,6]]}},"alternative-id":["S0963548300000560"],"URL":"https:\/\/doi.org\/10.1017\/s0963548300000560","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,6]]}}}