Tail bounds for the height and width of a random tree with a given degree sequence
Bibliographic record
Abstract
Abstract Fix a sequence c = (c1,…,cn) 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 v1,…,vn so that for each 1 ≤ i ≤ n, vi has exactly ci 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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.025 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.007 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".