Counting phylogenetic networks of level 1 and 2

Area of Science:

• Evolutionary biology
• Computational biology
• Graph theory

Background:

• Phylogenetic trees model evolutionary history but struggle with genetic material transfer.
• Phylogenetic networks generalize trees to account for reticulate evolution.
• Level-k networks are a specific class of phylogenetic networks.

Purpose of the Study:

• To provide enumeration formulas for rooted and unrooted level-1 and level-2 phylogenetic networks.
• To analyze the asymptotic distribution of network parameters, such as the number of cycles.
• To establish a foundation for studying complex evolutionary histories.

Main Methods:

• Developing a recursive description (combinatorial specification) for level-k phylogenetic networks.
• Applying enumerative, symbolic, and analytic combinatorics.
• Proving asymptotic normality for network parameter distributions.

Main Results:

• Exact and asymptotic enumeration formulas for level-1 and level-2 phylogenetic networks with a specified number of leaves.
• Proof of asymptotically normal distribution for parameters like the number of cycles.
• Established combinatorial specifications for these network classes.

Conclusions:

• The study provides a quantitative framework for understanding the complexity of phylogenetic networks.
• The combinatorial methods offer tools for analyzing evolutionary processes involving gene transfer.
• Future research can extend these methods to higher-order level-k networks.