Polynomial encoding of rooted trees with branch lengths

Area of Science:

• Evolutionary biology
• Computational phylogenetics
• Epidemiology

Background:

• Phylogenetic trees are crucial for evolutionary and epidemiological studies, requiring methods to quantify differences.
• Existing graph-polynomial encodings for tree comparison lack direct incorporation of branch lengths.
• Auxiliary structures are currently needed to analyze rooted trees with branch lengths using polynomial methods.

Purpose of the Study:

• To introduce a novel bivariate polynomial encoding for rooted trees with branch lengths.
• To enable direct incorporation of branch lengths into tree comparison using a recursive computation.
• To provide an accurate and computationally efficient method for analyzing phylogenetic tree differences.

Main Methods:

• Developed a bivariate polynomial encoding incorporating branch lengths recursively from leaves to the root.
• Proved the polynomial's uniqueness for rooted trees with branch lengths and no degree-two vertices.
• Applied the encoding to HIV-1 phylogenies from different epidemiological settings.

Main Results:

• The bivariate polynomial encoding directly integrates branch lengths into tree comparison.
• The method establishes a one-to-one correspondence between tree polynomials and trees with identical topology and branch lengths.
• The encoding successfully differentiated HIV-1 datasets, outperforming prior polynomial methods.

Conclusions:

• The new bivariate polynomial encoding is a powerful tool for comparing rooted phylogenetic trees with branch lengths.
• This method enhances the analysis of evolutionary and epidemiological data by accurately capturing tree topology and divergence times.
• The approach offers improved accuracy and efficiency for phylogenetic comparisons.