I want to know what will kill this tree.

Tordon. Cannot be bought at places like Home Depot but can find it on Amazon and farm stores like Tractor Supply. Break a stem and spray it, but carefully. Wear gloves and don't do it on a windy day.

If its small enough, pull it up. The tap root grows sideways.

If you use Tordon, be aware that it can "flash" into the surrounding soil. Meaning, the stuff is taken into the target tree tissue, carried into the root system, and can exit the root system. Then be taken up by neighboring plants and damage them, too.

Also, don't get it on the soil or into water. The stuff moves.

There is good reason we don't use it in forestry management any longer.

Sometimes Glyphosate (Roundup concentrate) is painted on freshly cut cambium tissue of woody plants. I've used that on my neighbor's invading bamboo and it works. I've never tried it on a tree, but have heard it works OK.

Glyphosate, when in contact with soil, decomposes quickly and therefore is not considered as hazardous as spilled or flashed Tordon.

I've successfully killed Trees of Heaven with Glyphosate. Late spring, cut slashes in the tree trunk (deep enough to reach the cambium) all the way around the trunk (but don't girdle it) and fill the slashes with undiluted roundup. Do that once a month. The tree will be dead by fall.

Thank you very much for your replies. It seems that glyphosate is safer than tordon and since my gardening skills are less than expert I will go with the glyphosate. ALSO - I do not have Trees yet. I have for the last 2 years had 4-6 foot seedlings that are thinner than the roots. The roots I have found travel and are rope-like. I plan to chop off the seedling and paint the stump. Does that sound right?

Yes, good plan.

Be sure you paint the cut seedlings and stems within 5 minutes of making the cuts, the sooner the better. If you wait, they'll callous over and won't take the poison down into the roots, which is where you want it. I put some glyphosate in small nail polish bottle with a brush inside that I got off Amazon so I can cut a few, pull it out of my pocket and paint the cuts, go on to next ones until done.

If the trees are only 4 or 5 ft, you may be able to pull them up. The taproot grows sideways (the better to sucker with) so if you pull the right direction, they just peel right out of the ground.

If they don't pull, treat the cut stump, wait a couple weeks then dig it up. Don't bother to treat them until they are in active growth in early summer.