No need to kill trees, as what you are looking for is the thinner, outer bark anyway.
Even if you were to cut a big thick piece off the tree, the outer "paper" you are collecting is not any easier to remove than if you Just spent your time collecting paper from live trees. Find a stand of similar, younger trees to collect from, as you will have to match thickness and color from many smaller pieces and an older tree tends to have knarlier bark.