Beavers!

Beavers could cut down trees and would move one log at a time to the closest river(the log would appear to be in it's mouth) and could pick up sticks. They would move the sticks to the closest river and place them there but as soon as it is placed it would become the block Beaver Dam. If you broke the Beaver Dam you would get 1 to 3 sticks or if you had Silk Touch on the tool you are using you could craft the beaver dam into 2 sticks. Beavers would spawn next to a river.