What's the best way to remove a small tree from my patio?

If there is room, cut the tip off the top of the tree, then spray stump/brush killer on it (or kerosene, but thats illegal)

Slicing the top will allow the stump killer to ease into the tree and kill it. The cameo layer under the bark is what you are trying to kill. That feeds the tree. But soak it in the stump killer anyway.

Also, if you take a drill and drill a bunch of holes into it, it will cause it to rot quicker. If you lived nearby he would do it for you granted you buy the supplies. Send me a message, we know people everywhere. I dont post personal info in asnwers :) But will be glad to help!!

Get a small saw or serrated knife. Cut into the bark as far as you can and all the way around the trunk as low as you can. This is called girding and will kill the tree fairly quickly. You will be left with the stump. Cut an X into the top of the trunk. This will speed up rotting. You could also try to chip out pieces of the stump or try digging it out, being careful not to crack the cement.

Cut it off right down to the cement. Trying to pull it out may crack the cement. Do you have a solid cement patio or a paver? If you have a pave style you can remove a few and pull the tree out. If one solid area, cut it off at the base and with a sharp, heavy knife cut down and remove as much as you can. You don't want to have a stub to trip over. A small, wood chisel would work fine also. Set something over this spot and monitor to see if it grows any. The roots may die out.

Cut the tree off close to the ground and put some Diesel fuel on the cut while it's still fresh.

Dig around it. Get a root cutter tool (its a chopping tool) from your local hardware store, chop the roots and pull it out.

Good luck!