If you cut the tree and treat the inner bark with a very concentrated treatment of glyphosate, it will go into the roots and kill the roots.
Same treatment for tree of heaven.
Tree of Heaven (Ailanthus altissima) - Identification & Control (without causing it to sucker!)
https://www.youtube.com/watch?v=hHsSCn6L450
I lived in Jerome, AZ for a while, and tree of heaven (called paradise tree around there) is everywhere. It destroyed a lot of older sidewalks and building foundations, especially while Jerome was a semi-ghost town in the 60s and 70s.
GREAT MINDS - !
I read yours after I wrote mine!!