I see no problem with burying dead animals under trees. I've done it plenty of times in the past.
From a
permaculture point of view, you may not wish to bury chemically euthanized animals near food trees. Sodium pentobarbital is the primary drug used fur euthanasia, but others including various tranquilizers, ketamine, and others are frequently used. I'm not sure about the others, but sodium pentobarbital is rather persistent in the environment. In the past, I used to bury euthanize animals under pine trees. The trees grew absolutely great.