let's work with dihybrid crosses dihybrid crosses are like monohybrid crosses but instead of one trait they involve two traits metal track two traits the color which could either be green or yellow and the texture which could be either round or wrinkled yellow is dominant while green is recessive round is dominant while wrinkled is recessive in this cross we are taking a dihybrid individual which means that this individual is heterozygous both for the texture trait as well as the color trait and crossing them to another dihybrid individual hence the name the dihybrid cross now one approach we could take to computing the genotypic proportions here is the old familiar punnett squares and in order to do punnett squares first we have to imagine meiosis and write down all the different gametes this individual can produce well you could have a big r big y gammy or you could have a little r big y gamey or you could have a big r little y gamete and finally you can have a little r little y gamete furthermore from mendel's law of independent assortment each of these gametes are equally likely and therefore we will have a quarter big our big y a quarter little r big y a quarter big r little y and a quarter little r little y and the other individual since both the individuals involved in this cross have the same genotype they will produce the same type of gametes and we can go ahead and make a punnett square which will have 16 squares 16 cells and go and use the product rule to compute the genotypic proportions for example in this first cell resulting from the fertilization of a big r big y egg with a big r big y sperm i will get at a frequency of 1 16 a big r big r big y big y zygote now this kind of competition although it's certainly possible is quite tedious since you have to do to compute 16 genotypes and 16 frequencies and the the number of possibilities grows with the number of chromosomes that that are under consideration so if we were doing in for instance a trihybrid cross with tracking three traits on three different chromosomes you would end up with eight different types of gametes and a punnett square with 64 cells in it the other issue is that we rarely if ever need all the different genotypes usually we are trying to determine the frequency of a specific genotype in in the progeny so this is too complicated instead we're going to use branching diagrams to radically simplify how we compute genotypic proportions phenotypic proportions for dihybrid crosses and not only that branching diagrams scale much better as you increase the number of traits to 3 4 5 and so on the first thing to do is to use mendel's law of independent assortment to divide and conquer since these two traits the color of the piece as well as the texture of the piece are on separate chromosomes they assort independently and that means that we can consider each trait independently and this dihybrid cross is actually a combination of two monohybrid crosses the first monohybrid cross is between the head for the round versus wrinkled trait and the other monohybrid cross is between the yellow and green traits or monohybrids for the yellow and green traits and we know that this cross yields three genotypes a quarter are big r big r a half our big r little r the heads and a quarter r little r little r the same is true for the the color trade the yellow versus green trade and you can get a quarter big y big y a half big y little y and a quarter little y little y when we're looking at the dihybrid rate however we will get all the possible combinations between the two traits for example you could have big r big r for the texture trait and then be big y big y for the color tray you could be big r big r for the texture trade and big y little y for the color tray and so on and we can use the branching diagram to represent all of the possible combinations between the genotypes of the texture trait and the genotypes of the color trait next we will compute using this branching diagram the frequencies of the different genotypes just making some space here now we can follow the branches of this tree to write down the different genotypes that are possible in the progeny of this dihybrid cross as well as their frequencies for example this branch corresponds to the genotype big r big r and big y big y and we can compute the frequency of this genotype using the product rule since um you know in order to be big r big r and big y big y you must inherit both the big r chromosomes and the big y chromosomes and the inheritance of these two traits is independent by mendel's law of independent assortment that means i can simply multiply the probabilities of the two events to obtain the probability that the two events will occur together and therefore the probability of the big r big r big y big y genotype is 1 16. let's do another example what is the frequency of the genotype where the first trait the the texture round versus wrinkled is heterozygous and the second trait which is the color is homozygous for the dominant allele big y big y and all we need to do is to multiply the probabilities of getting a head for the first trait which is half with the probability of getting a homozygous individual for the the dominantly the big y allele and that's half times a quarter and so the probability of getting an individual who is big r over little r and big y over big y is 1 8. i'm leaving writing down all the remaining genotypes as well as computing their frequencies using the product rule as an exercise and so we have seen that two things the first mendel's law of independent assortment that allows us to divide and conquer as well as branching diagrams that allow us to evaluate all the possible combinations between the genotypes of the two traits are very useful tools to compute the the the genotypes as well as their proportions and we will be making extensive use of these tools in the future you