Blow up a round balloon. Make sure to get balloons that remain spherical when you blow them up; many shapes are available at party supply stores. When inflated, the balloon should be about the size of a man's fist.
Curl the end of a piece of wire into a circle about as large as a quarter. Eighteen or 20 gauge wire should be soft enough to curl with your fingers but still hold its shape. Slip your circle over the balloon's knot.
Gently begin bending the wire around the balloon in different directions. Be patient and bend slowly so you don't pop the balloon accidentally. Keep wrapping until you achieve a mostly solid sphere that looks a bit like asymmetrical mesh.
Use a needle to pop the balloon and discard the shards. Snip lengths of twine to hang your spheres from tree branches.