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Bibliographic record
Abstract
Random binary search trees are obtained by recursively inserting the elements σ(1),σ(2),…,σ(n) of a uniformly random permutation σ of [n]={1,…,n} into a binary search tree data structure. Devroye (J. Assoc. Comput. Mach. 33 (1986) 489–498) proved that the height of such trees is asymptotically of order c∗logn, where c∗=4.311… is the unique solution of clog((2e)/c)=1 with c≥2. In this paper, we study the structure of binary search trees Tn,q built from Mallows permutations. A Mallows(q) permutation is a random permutation of [n]={1,…,n} whose probability is proportional to qInv(σ), where Inv(σ)=|{i<j:σ(i)>σ(j)}|. This model generalizes random binary search trees, since Mallows(q) permutations with q=1 are uniformly distributed. The laws of Tn,q and Tn,q−1 are related by a simple symmetry (switching the roles of the left and right children), so it suffices to restrict our attention to q≤1. We show that, for q∈[0,1], the height of Tn,q is asymptotically (1+o(1))(c∗logn+n(1−q)) in probability. This yields three regimes of behaviour for the height of Tn,q, depending on whether n(1−q)/logn tends to zero, tends to infinity or remains bounded away from zero and infinity. In particular, when n(1−q)/logn tends to zero, the height of Tn,q is asymptotically of order c∗logn, like it is for random binary search trees. Finally, when n(1−q)/logn tends to infinity, we prove stronger tail bounds and distributional limit theorems for the height of Tn,q.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it