Jim, yeast can reproduce two different ways. The first way that a yeast cell can reproduce is by budding a new daughter cell. The scientific term for this type of reproduction is mitosis. Mitosis is asexual reproduction. The second type of reproduction is meiosis, or sexual reproduction. During meiosis, a cell doubles its DNA and then splits the doubled DNA between four spores, each of which contains half of the mother cell's DNA. These spores are known as haploid cells they have half of the mother cell's chromosomes. The four haploid cells are divided into two a cells and two α (the Greek letter for alpha) halpoid cells. The a and α cells are basically different sexes, which means that an a cell can mate with an α cell, forming a new diploid cell, basically, the same process that happens when a sperm cell (haploid cell) fertilizes an egg (haploid cell).
Meiosis in brewing strains is rare due to the polyploid nature of brewing yeast. Many strains are triploids. Triploid strains are basically sterile due to the difficulty of undergoing meiosis. Tetraploid strains have an even number of sets of chromosomes, but many brewing strains are not perfect tetraploids. They are what are known as aneuploids. Aneuploidy in yeast means that the total number of chromsomes is not evenly divisible by 16. The strains within the Frohberg (allotetraploid) and Saaz (allotriploid) families of lager yeast that have been sequenced all exhibit aneuploidy.
The lack of genetic diversity in lager yeast is due to fact that Frohberg and Saaz families are from one or two hybridization events (the number of hybridization events is a debated topic). Many modern scientists question brewing scientists did not inadvertently discard other hybrids during the early days of pure cultures. The differences in lager strains today are the result of selective pressure