Biologist here, the body itself isn't what grows the "greener grass". It's just the start of a long biological process that will lead to ecological growth.

A tree isn't going to feed directly off of the body. But the decay process will provide nutrients to the tree. We're talking about insects and fungus at various levels of the process here. You can look up things like the Trophic Levels and Nutrient Cycles for more details on this whole process.

TLDR would be that the corpse floods the area with nutrients and maybe even kills off the plants with over abundance of nitrogen, but then fungus and bugs move in, then bigger bugs and small animals, and so on and then better plant systems. It's kind of neat.