Does the bit of the leek you eat go down from the surface, with fine roots growing sideways from it, like carrots, turnips and parsnips or does it grow up from the surface, until someone adds more soil around it to blanch it like celery, with fine roots growing down wards from the flat base?
It is obvious that celery sticks are stems, not roots, and that carrots etc are tap roots. Leeks are nor so obvious.
Celery has joints part way up the stems, and leaves at the top of the stems. Leeks have joints too, which (to me) means that the lower parts are stems and the top parts are leaves. Of course, if you always buy both celery and leeks "trimmed", for convenience, you only get the neat lower parts and never see the leaves that were at the top of the stems.
Answer - leeks are like celery. You eat the stems. You don't eat the roots, which you cut off from the base. You trim off the outer, coarse leaves from the top, but sometimes you cook the softer green top leaves from inside.