The Running Bowline
A bowline is a loop that won't slip. A running bowline has a bowline tied in the end of the rope with the looped end placed over or around an object like a tree branch. The other end of the rope is passed through the loop and drawn tight.
The bowline knot is easy to tie. Make a single circle in the rope where you want the knot to be. Put the end of the rope through the circle, around the rope and back through the circle. Tighten the knot so it looks neat and you can see the loop in the knot. Neatness counts with this knot, and it won't slip if tied correctly.
Half Hitches
A half hitch is a simple knot, yet it has very good holding strength. It's useful for attaching a rope to a tire for a tire swing, or for a wooden seat with holes in it for the rope. A single half hitch isn't good for much and can easily slip. Two or three hitches have better holding strength, and the more tension or weight put on the knot, the tighter it gets.
The half hitch is very simple. Make a half hitch by placing the rope around an object or another rope. Pass the end of the rope around the other side of the line, then put it through the loop between the object and the knot. Draw the knot up tight to the object and add additional half hitches for additional strength.
Stopper Knot
The stopper knot, or Ashley's stopper knot, is a large knot placed on the end of a rope to prevent it from pulling through a hole. You could use it on the bottom of a wooden seat after tying the rope to the tree branch or play set.
Tie a stopper knot by turning a circle in the end of the rope, then make a second circle that overlaps the first. Pull the first circle through the second, then take the end of the rope across the front of the knot. Pass it between the two loops by putting it over the first loop and under the second loop. Bring the end up and through the top of the first loop from behind. Pull the knot tight.
Cow Hitch
The cow hitch has an advantage over the running bowline. Once it's tied and tight on a tree branch, it won't rub back and forth. The rope around the branch stays put, even when the swing is moving back and forth. The cow hitch is a variant of the half hitch, but has superior holding strength and doesn't slip easily. The disadvantage is you have to get up to the tree branch to tie it.
Pass the rope over the tree branch and tie a half hitch in it. Pass the end of the rope around the front of the rope to make a second loop -- the half hitch makes the first -- then under the tree branch, and bring it back up over the top of the branch. Put the rope through the two loops and pull the knot tight onto the bottom of the branch. Make sure the two passes around the tree branch don't cross each other and are tight around the branch.