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A Simple Protocol for Mapping the Plant Root System Architecture Traits

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On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees.

Noah A Rosenberg¹

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.

Keywords:

05C05 92B10 92D15 Phylogenetics quadratic recursion unlabeled trees

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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.