Reconstructing Unrooted Phylogenetic Trees from Symbolic Ternary Metrics.

Area of Science:

• Graph theory
• Combinatorics
• Computational biology

Background:

• Böcker and Dress established a correspondence between rooted trees and ultrametrics.
• The current study extends this to unrooted trees and vertex coloring.

Purpose of the Study:

• To characterize ternary maps derived from colored unrooted trees.
• To reconstruct unrooted trees and their colorings from these maps.
• To identify conditions for binary trees and develop reconstruction algorithms.

Main Methods:

• Mapping triples of leaves to median vertex colors.
• Characterizing ternary maps using 4- and 5-point conditions.
• Developing a bottom-up reconstruction algorithm for general trees.

Main Results:

• Characterization of valid ternary maps for unrooted colored trees.
• Demonstration of tree and coloring reconstruction from valid ternary maps.
• Identification of a condition distinguishing binary trees.

Conclusions:

• Ternary maps provide a robust framework for studying unrooted colored trees.
• The developed conditions and algorithms enable tree reconstruction.
• This work offers new insights into tree structures and their representations.