Generalizing Matrix Representations to Fully Heterochronous Ranked Tree Shapes

Area of Science:

• Evolutionary biology
• Computational phylogenetics
• Combinatorics

Background:

• Phylogenetic tree shapes encode evolutionary history.
• Existing F-matrix framework applies to isochronous trees (uniform leaf sampling time).
• Rooted phylograms with branch lengths as evolutionary distance are common but heterochronous.

Purpose of the Study:

• Extend the F-matrix framework to fully heterochronous ranked tree shapes.
• Establish a bijection between F-matrices and these tree shapes.
• Develop probabilistic models on ranked tree shapes.

Main Methods:

• Discretizing time in rooted phylograms to create fully heterochronous ranked tree shapes.
• Establishing an explicit bijection between a class of F-matrices and these tree shapes.
• Utilizing matrix properties for enumeration and probabilistic modeling.

Main Results:

• A new bijection is established between F-matrices and fully heterochronous ranked tree shapes.
• The matrix representation allows for straightforward enumeration of valid tree shapes.
• Probabilistic models on ranked tree shapes are developed.

Conclusions:

• The F-matrix framework is successfully extended to heterochronous phylogenetic trees.
• This provides a powerful combinatorial tool for analyzing evolutionary relationships.
• The work opens new avenues for probabilistic modeling in phylogenetics.