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    We introduce the diameter as a mathematical property to assess species trees by analyzing deep coalescence (DC) costs. Our findings provide efficient algorithms for computing diameters, improving the accuracy of DC cost assessments.

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    Area of Science:

    • Computational Biology
    • Phylogenetics
    • Evolutionary Biology

    Background:

    • The minimizing deep coalescence (MDC) problem aims to find a species tree that minimizes deep coalescence (DC) events.
    • Assessing the accuracy of species trees is crucial for understanding evolutionary history.
    • The DC cost quantifies the discordance between gene trees and species trees due to deep coalescence.

    Purpose of the Study:

    • To investigate the mathematical properties of the diameter of gene trees and species trees.
    • To develop efficient algorithms for computing these diameters.
    • To enhance the assessment of species trees in the context of the MDC problem.

    Main Methods:

    • Mathematical analysis of DC cost and tree diameters.
    • Development of algorithms for computing diameters under bijective and general leaf labelings.
    • Experimental evaluation of computed diameters against naive upper bounds.

    Main Results:

    • Fundamental mathematical properties of tree diameters were proven.
    • Efficient algorithms, including a linear-time algorithm for the bijective case, were developed.
    • Experimental results show diameters grow slower than expected, validating the approach.

    Conclusions:

    • The diameter is a valuable metric for assessing species trees and DC costs.
    • The developed algorithms provide efficient and accurate computation of tree diameters.
    • This work offers improved methods for evaluating species tree accuracy in phylogenetics.