If you have a backyard tree just begging for a swing you can easily make a small child swing out of a 18L bucket using a design from environmental weblog EcoProjecteer.
The only materials you'll need for the project are the bucket and a rope, but you'll also need a drill, a saw, sandpaper and a permanent marker. Mark and cut out two leg holes, drill holes for the rope and mark and cut the sides of the bucket. Spend a few minutes sanding down any rough surfaces then take the swing outside and tie one of the rope to a sturdy tree branch, run it through the holes in the bucket and tie the other end to the same tree branch.
Not only will this keep a bucket out of a landfill but since most child swings cost $US25-50 you'll save some cash as well. This should work great for kids from around one year old until age three or four; by then they'll be ready to graduate to a tire swing.