{"id":"W4415782818","doi":"10.1007/s11538-025-01538-7","title":"Tree Height and the Asymptotic Mean of the Colijn–Plazzotta Rank of Unlabeled Binary Rooted Trees","year":2025,"lang":"en","type":"article","venue":"Bulletin of Mathematical Biology","topic":"Stochastic processes and statistical mechanics","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"McGill University","funders":"National Human Genome Research Institute; Natural Sciences and Engineering Research Council of Canada; Vetenskapsrådet; National Institutes of Health; National Science Foundation","keywords":"Random binary tree; Binary tree; Random tree; Rank (graph theory); Tree (set theory); Binary number; Optimal binary search tree; Binary search tree; Bijection","routes":{"ca_aff":true,"ca_fund":true,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0008779425,0.0001935911,0.0009057375,0.00006001116,0.00006660841,0.000005073409,0.0004849828,0.0001789817,0.0003485008],"category_scores_gemma":[0.007145637,0.00009570003,0.0001535681,0.0002239156,0.001113831,0.000004772795,0.0002930602,0.0001681926,0.000005238814],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.000008606372,"about_ca_system_score_gemma":0.00005674727,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00001336856,"about_ca_topic_score_gemma":0.000008162403,"domain_scores_codex":[0.9981911,0.0002889292,0.0008960046,0.0002155356,0.0001825127,0.0002259],"domain_scores_gemma":[0.9897945,0.009122333,0.0003894838,0.0004817016,0.0001718614,0.00004011408],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.0001868414,0.0002573045,0.00001969904,0.000930679,0.0001485858,6.019662e-7,0.0002059791,6.338672e-7,0.001332724,0.9942603,0.001530117,0.001126516],"study_design_scores_gemma":[0.002321603,0.0002444064,0.0001406452,0.0004595389,0.0003187762,0.000009253222,0.0002163699,0.002468756,0.003952925,0.9893782,0.0003878573,0.0001016661],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.3832617,0.003718293,0.5196219,0.03371983,0.0006556059,0.004801203,0.0003630155,0.0001387534,0.05371977],"genre_scores_gemma":[0.9863048,0.00004207861,0.01288349,0.000140177,0.00001394067,0.00003782291,0.000001868159,0.00001489353,0.0005609966],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.6030431,"threshold_uncertainty_score":0.8554511,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.01538501404705447,"score_gpt":0.2773529840010202,"score_spread":0.2619679699539657,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}