Here is the full list of all 16 Desktop Tree achievements.
Catch 100 bugs.
Grow a tree and harvest its seeds.
Your tree is over 30 days old.
Your tree is over 7 days old.
Your tree is almost as tall as you, have you considered shearing it?
Flowers become fruits.
Your tree blooms!
Your tree has over 100 leaves, have you counted?
Catch a bug.
You have found the reset button.
Grow a second tree and harvest its seeds.
Grow a third tree and harvest its seeds.
Grow 5 trees and harvest their seeds.
Cut off the whole canopy.
Cut a branch.
Catch 1000 bugs.