Final answer:
Phylogenetically independent contrasts are important in scenarios where we want to eliminate the effects of shared ancestry and understand the relationships between traits or variables.
Step-by-step explanation:
Phylogenetically independent contrasts are important in all of the given scenarios: 1) to infer whether variation in beak size is heritable in a single population, 2) to test for a correlation between diet and toxicity across populations within a species, and 3) to test for a correlation between genome size and host specificity among an array of parasitic species. In each scenario, phylogenetically independent contrasts help eliminate the effects of shared ancestry and better understand the relationships between traits or variables.