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On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees
1Department of Biology, Stanford University, Stanford, CA 94305 USA.
Discrete Applied Mathematics (Amsterdam, Netherlands : 1988)
|December 28, 2020
Summary
This study maps unlabeled binary rooted trees to integers, revealing that the smallest and largest ranks for trees with n leaves grow at different rates. This difference impacts how tree structures change with increasing integer ranks.
Area of Science:
- Computational Biology
- Phylogenetics
- Discrete Mathematics
Background:
- A bijective scheme by Colijn & Plazzotta (2018) associates unlabeled binary rooted trees with positive integers.
- This scheme assigns ranks recursively based on the ranks of left and right subtrees.
Purpose of the Study:
- To determine recursions for the smallest (a_n) and largest (b_n) ranks assigned to trees with n leaves.
- To analyze the growth rates of a_n and b_n and their implications for tree structure distribution.
Main Methods:
- Derivation of recursive formulas for a_n and b_n based on the Colijn & Plazzotta ranking scheme.
- Analysis of the asymptotic behavior of a_n and b_n as the number of leaves (n) increases.
- Identification of tree structures corresponding to a_n (maximally balanced) and b_n (caterpillar).
Main Results:
- The smallest rank (a_n) for n leaves grows exponentially, bounded by a_n < 1.5^n, with a specific growth rate for powers of 2.
- The largest rank (b_n) for n leaves also grows exponentially, but at a different rate (β ≈ 1.05653).
- The significant difference in growth rates between a_n and b_n indicates a wide range of leaf numbers for trees with increasing ranks.
Conclusions:
- The integer ranking of trees reveals a non-uniform distribution of leaf numbers across ranks.
- This finding has implications for understanding tree structure diversity in applications like evolutionary biology.
- The study provides a mathematical framework for analyzing the relationship between tree topology and rank.
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