This is a most unfortunate situation and the reason why one has to be cautious about willows. From the sounds of it, now that you've removed the tree, the outlying (detached) root system is sending off shoots to take over a new tree. The removal of the main tree had been a mistake. Ideally, you;d chop it down to a stump and apply round-up heavily and let the poison travel down it's root system and this will kill eventuall kill off the entire tree.

Now in trying eradicate the roots, how have you been digging them up? Cutting them where you find them? This leads to more new plants developing where you've cut off the roots. I cannot think of any ways to go about it but to see if you can find each and every shoot, apply Round Up heavily and let the toxin travel through it's root system. When there is evidence that the shoots are dying, then dig the root system out and be careful not to cut them and accidentally create more plants. Sadly this may cause you to dig up your lawn. As for plants growing through the deck, if you can find a way to crawl beneath and install a weed cloth to cut off it's access to sunlight and water, that would kill off any new growth.

Oh man... does that suck. Wish we had known to spray down the root ball instead of digging it up for what you say makes sense.

We pick and pick and pick the new shoots on a weekly basis which are now all over the yard. I will try the round up on them instead of pulling. If this doesn't work then I suppose we are left with the task of pulling up the sod next spring. Boo.... :(