Final answer:
The minimum number of nodes in a binary tree with a maximum depth of 4 is 5, creating a linear path from root to leaf. In genetics, for a gene with four alleles, there are 10 possible genotypes due to the combinations of heterozygous and homozygous pairings.
Step-by-step explanation:
If a binary tree has a max depth of 4, the minimum number of nodes is 5. This is because the minimum number of nodes in a binary tree of any depth can be represented by having just one child at each level. This creates the longest possible path from the root to a leaf without filling out any other breadth of the tree. Therefore, you start with a single root node, and then at each subsequent level (up to 4), you add one more node to continue the single path, resulting in a total of 5 nodes.
When a population has a gene with four alleles, the number of possible genotypes can be calculated using the combination formula for a set of unique items taken two at a time which includes heterozygous and homozygous combinations. The formula is n(n+1)/2, where n represents the number of different alleles. Substituting n=4 into this formula, 4(4+1)/2 = 10, means there are 10 possible genotypes for a gene with four alleles circulating in the population.