Unlike people, trees do not have cd annuities; that means you would not get any benefits from the trees once they are gone. Trees are more beneficial to human kind when they are alive, thus we need to plant and take care of trees.
You may grow trees in your lawn area. However,trees appear to be very peaceful coexisting with other trees but the truth is they weaken each others immune system. Thus, you need to do something to take care of them and observe any unusual changes.
You can also plant trees in an organic yard. Pesticides and insecticides are not only harmful for humans and environment; they are harmful for trees and plants too.
If you are really serious about growing trees, you need a better understanding about the kind of soil that the trees required. The type of climate and many other things that will help you grow healthy trees in your own lawn or yard.