I thought it was so cool that a little tree had volunteered in our yard. By the time I figured out it was a weedy tree of heaven, man what a job it was to dig out that long root!
Oh, how those weed trees hang onto life! Here are (left) Siberian elm (Ulmus pumila) and tree of heaven (right) saplings. The elm may be two years old. The tree of heaven is at least two -- see where it was cut off in year one, and how it came back two-for-one?
A tree's tap root eventually dwindles, as roots at the edge of the dripline out-perform it in gathering water and nutrients. But in the first couple of years the tap root is a formidable barrier to easy removal!
Compare the bud scar (where the petiole was attached last year) and winter bud of this red horsechestnut (Aesculus x carnea) to those of the tree of heaven (photo at the top of this page). You can distiguish between them, right? Then you'll know to pull out the tree of heaven, right away!