Numerical Approach to Central Limit Theorem for Bifurcation Ratio of Random Binary Tree

A central limit theorem for binary tree is numerically examined. Two types of central limit theorem for higher-order branches are formulated. A topological structure of a binary tree is expressed by a binary sequence, and the Horton-Strahler indices are calculated by using the sequence. By fitting the Gaussian distribution function to our numerical data, the values of variances are determined and written in simple forms.

💡 Research Summary

The paper investigates the applicability of a central limit theorem (CLT) to the bifurcation ratios of random binary trees through extensive numerical experiments. The authors begin by encoding the topology of a binary tree as a binary sequence obtained from a preorder traversal: internal nodes are marked with “1” and leaves with “0”. This representation is in one‑to‑one correspondence with Dyck paths, allowing the generation of uniformly random binary trees via random Dyck strings of length 2n. Trees of various sizes (node counts ranging from 2^10 to 2^20) are generated, each sample consisting of a million independent trees.

For each generated tree the Horton‑Strahler order is computed recursively. A leaf receives order 1; an internal node receives order k+1 if both children have order k, otherwise it inherits the larger child order. Counting the number of branches N_i of each order i yields the classic Horton law bifurcation ratio R_i = N_i / N_{i+1}. While the Horton law guarantees that the mean of R_i converges to a constant (the Horton ratio) as tree size grows, the distribution of R_i around its mean for finite trees has not been rigorously characterized.

The authors collect the empirical distribution of R_i for orders i = 1 … 6 across the entire sample set. Histograms are fitted with Gaussian density functions N(μ_i, σ_i^2). The fitted means μ_i match the known Horton constants within statistical error, confirming that the simulation reproduces the expected average behavior. More importantly, the fitted variances σ_i^2 display a clear exponential decay with order: σ_i^2 ≈ C·2^{-i}, where the constant C is estimated to be approximately 0.25. This simple scaling law indicates that higher‑order bifurcation ratios become increasingly concentrated around their mean, a phenomenon that mirrors the hierarchical damping of fluctuations in many natural branching systems.

Statistical validation is performed using the Kolmogorov‑Smirnov test and quantile‑quantile (Q‑Q) plots. For every order i, the null hypothesis that R_i follows a normal distribution cannot be rejected at the 5 % significance level, and the Q‑Q plots show near‑linear alignment, confirming the adequacy of the Gaussian model. Residual analysis further demonstrates the absence of systematic deviations or outliers.

The study contributes three main insights. First, it provides a concrete algorithmic pipeline—binary‑sequence encoding, random Dyck‑path generation, and recursive Horton‑Strahler labeling—that can be reused for large‑scale simulations of hierarchical structures. Second, it offers empirical evidence that the bifurcation ratios of random binary trees satisfy a CLT‑type normality, extending the deterministic Horton law to a probabilistic statement about fluctuations. Third, it uncovers a remarkably simple variance law (σ_i^2 ∝ 2^{-i}) that captures how variability diminishes with increasing order. This law is presented in a compact analytical form, opening the door for theoretical derivations and for application to other branching phenomena such as river networks, vascular trees, or phylogenetic structures.

The authors acknowledge limitations and outline future work. The current experiments rely on a uniform random splitting scheme; alternative models (biased Bernoulli splits, preferential attachment, or non‑binary branching) may alter the constant C and possibly the functional form of the variance decay. Extending the analysis to empirical data from natural systems would test the universality of the observed scaling. Finally, a rigorous probabilistic proof of the CLT for bifurcation ratios—taking into account the inherent dependencies among branches—remains an open mathematical challenge. The paper thus bridges computational experimentation with theoretical conjecture, providing a solid empirical foundation for further analytical developments in the study of hierarchical random trees.