Tree Height and the Asymptotic Mean of the Colijn–Plazzotta Rank of Unlabeled Binary Rooted Trees
Bibliographic record
Abstract
Abstract The Colijn–Plazzotta ranking is a bijective encoding of the unlabeled binary rooted trees with positive integers. We show that the rank f ( t ) of a tree t is closely related to its height h , the maximal path length from a leaf to the root. We consider the rank $$f(\tau _n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of a random n -leaf tree $$\tau _n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> under each of three models: (i) uniformly random unlabeled unordered binary rooted trees, or unlabeled topologies; (ii) uniformly random leaf-labeled binary trees, or labeled topologies under the uniform model; and (iii) random binary search trees, or labeled topologies under the Yule–Harding model. Relying on the close relationship between tree rank and tree height, we obtain results concerning the asymptotic properties of $$\log \log f(\tau _n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>log</mml:mo> <mml:mo>log</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In particular, we find $${\mathbb {E}}\{\log _2 \log f(\tau _n)\} \sim 2 \sqrt{\pi n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mo>log</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>log</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>∼</mml:mo> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mrow> <mml:mi>π</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math> for uniformly random unlabeled ordered binary rooted trees and uniformly random leaf-labeled binary trees, and for a constant $$\alpha \approx 4.31107$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>4.31107</mml:mn> </mml:mrow> </mml:math> , $${\mathbb {E}}\{\log _2 \log f(\tau _n)\} \sim \alpha \log n $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>{</mml:mo> <mml:msub> <mml:mo>log</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>log</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>}</mml:mo> <mml:mo>∼</mml:mo> <mml:mi>α</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> for leaf-labeled binary trees under the Yule–Harding model. We show that the mean of $$f(\tau _n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> itself under the three models is largely determined by the rank $$c_{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>c</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> of the highest-ranked tree—the caterpillar—obtaining an asymptotic relationship with $$\pi _n c_{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mi>c</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> </j
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How this classification was reachedexpand
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 | 0.001 | 0.007 |
| 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 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".