I see people going through and petting the egg trees but not watering or harvesting. Well, I just got an egg from harvesting, so you might want to be a little more thorough. ;)
I harvested, petted and watered a ridiculous amount. Even though I wasn't paying attention to which action got me the egg at the way beginning of the egg hunt, once I started paying attention, I never got any eggs from watering or harvesting even though i continued to do so anyways always. I got many many eggs from petting though.
I also only got eggs from trees by petting. Like Omnika, I didn't pay attention in the very beginning, but once I did, I only petted them. figured it saved me time and let others water in case they got eggs.
I initially thought you only got eggs from harvesting, but then later realized you got them from all three actions...sadly, I never got any eggs from the egg plants...only from squeezing chickens...though even there, I had a system in place (I went to the neighborhoods in Alakol and found streets with 7 or more chickens before initiating a gurly drink...which could get me through about 3 streets of chicken squeezing = 7x5x3 = around 105 squeezes per drink...)