In "A White Heron" what do Sylvia and the heron have in common?
Both Sylvia and the heron have quite a few things in common. Both are solitary, lone creatures that are most comfortable surrounded by nature and its beauty. The heron is a rare bird, just as Sylvia, a girl who loves the country life and living in solitude, is a rare type of person. Most people enjoy living in the hustle and bustle of the city, surrounded with friends and constant activity. Sylvia, however, does not, and feels at home in the pleasant environment of nature--so, she is, like the heron, rare in that regard.
Another way that Sylvia and the heron are similar is that they are mysterious and elusive to common human understanding. The young hunter figured he could offer Sylvia money and that would be more than enough to prompt her to help him hunt down the bird. This is a pretty logical assumption to make, as it was a lot of money, and she and her grandmother were not well-off, and for most people, it would have been enough. Not for Sylvia though; she possess a unique and mysterious perspective, and the hunter does not understand it. He can't grasp her inner maturity and viewpoint, and so does not succeed in finding the bird. The heron is the same way--hard to find, hard to study, hard to understand, and not as easily accessible as other birds.
I hope that those thoughts help a bit--good luck!