Can be traded.
+2 for climbing tree, +6 for retrieving egg, -6 for giving away egg.
Climb the tree in F7 by using an action on it (the hand). On the second screen, you will see the egg in the nest. Walk carefully across the limb to the nest and then you can pick up the egg.
The tree branch is a bit narrow and it isn't too difficult to fall off. However, falling off will not kill you, so take as many tries as you need.
The egg isn't needed anyway in the game—you can complete the game with or without it. Walking on the tree limb can be tricky, but your best bet is to make small steps staying near the center. If you fall off, just try again.