Fumio Tajima developed a mathematical theory to analyze the expected evolutionary relationships among DNA sequences in finite populations, assuming evolutionary change is driven by mutation and random genetic drift. The study investigates the statistical properties of nucleotide differences and heterozygosity, revealing that estimates of these parameters have significant variance, largely due to stochastic factors. Increasing sample size does not significantly reduce this variance. The distribution of sample allele frequencies is also studied, showing that a small sample size is sufficient to understand the distribution pattern. In populations with negligible recombination, such as mitochondrial DNA, evolutionary trees can be constructed. These trees show that nucleomorphs from different populations are often more similar than those from the same population. Three possible explanations for this are natural selection, recent gene migration, and stochastic genetic drift. However, there is no theoretical study on expected phylogenetic trees under these hypotheses. The paper presents a mathematical theory on the expected genealogy of nucleons and the number of nucleotide differences among them. It also investigates the expected distribution of sample nucleomorph frequencies. The study considers single and two populations, analyzing the topological relationships and branch lengths of nucleon genealogies. It also examines the number of nucleotide differences between randomly chosen nucleons, showing that the average number of differences is influenced by mutation rates and population size. The study also discusses nucleon diversity, defined as the probability that two randomly chosen nucleons are different. It uses the K-allele model and the infinite-allele model to analyze nucleon diversity and its variance. The study finds that the average number of nucleotide differences is more informative than heterozygosity or nucleon diversity for measuring genetic variation within populations. The study also examines the distribution of sample nucleomorph frequencies, showing that the expected number of nucleomorphs with a given frequency can be calculated. The study concludes that the average number of nucleotide differences is a reliable measure of genetic variation within populations, despite its large variance. The study also discusses the implications of linked loci and the correlation between them, showing that highly polymorphic loci are often linked to other highly polymorphic loci. The study provides a comprehensive analysis of the evolutionary relationships among DNA sequences in finite populations.Fumio Tajima developed a mathematical theory to analyze the expected evolutionary relationships among DNA sequences in finite populations, assuming evolutionary change is driven by mutation and random genetic drift. The study investigates the statistical properties of nucleotide differences and heterozygosity, revealing that estimates of these parameters have significant variance, largely due to stochastic factors. Increasing sample size does not significantly reduce this variance. The distribution of sample allele frequencies is also studied, showing that a small sample size is sufficient to understand the distribution pattern. In populations with negligible recombination, such as mitochondrial DNA, evolutionary trees can be constructed. These trees show that nucleomorphs from different populations are often more similar than those from the same population. Three possible explanations for this are natural selection, recent gene migration, and stochastic genetic drift. However, there is no theoretical study on expected phylogenetic trees under these hypotheses. The paper presents a mathematical theory on the expected genealogy of nucleons and the number of nucleotide differences among them. It also investigates the expected distribution of sample nucleomorph frequencies. The study considers single and two populations, analyzing the topological relationships and branch lengths of nucleon genealogies. It also examines the number of nucleotide differences between randomly chosen nucleons, showing that the average number of differences is influenced by mutation rates and population size. The study also discusses nucleon diversity, defined as the probability that two randomly chosen nucleons are different. It uses the K-allele model and the infinite-allele model to analyze nucleon diversity and its variance. The study finds that the average number of nucleotide differences is more informative than heterozygosity or nucleon diversity for measuring genetic variation within populations. The study also examines the distribution of sample nucleomorph frequencies, showing that the expected number of nucleomorphs with a given frequency can be calculated. The study concludes that the average number of nucleotide differences is a reliable measure of genetic variation within populations, despite its large variance. The study also discusses the implications of linked loci and the correlation between them, showing that highly polymorphic loci are often linked to other highly polymorphic loci. The study provides a comprehensive analysis of the evolutionary relationships among DNA sequences in finite populations.