Tail bounds for the height and width of a random tree with a given degree sequence
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Bibliographic record
Abstract
Abstract Fix a sequence c = ( c 1 ,…, c n ) of non‐negative integers with sum n − 1. We say a rooted tree T has child sequence c if it is possible to order the nodes of T as v 1 ,…, v n so that for each 1 ≤ i ≤ n , v i has exactly c i children. Let \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}${\mathcal T}$\end{document} be a plane tree drawn uniformly at random from among all plane trees with child sequence c . In this note we prove sub‐Gaussian tail bounds on the height (greatest depth of any node) and width (greatest number of nodes at any single depth) of \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}${\mathcal T}$\end{document} . These bounds are optimal up to the constant in the exponent when c satisfies \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}$\sum_{i=1}^n c_i^2=O(n)$\end{document} ; the latter can be viewed as a “finite variance” condition for the child sequence. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| 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