The roots of willow trees, like silver maples and poplars, can be difficult to control even after the tree has been cut down. Their shallow root system is a cache of stored nutrients that will continue to promote growth.
The roots of a willow tree can extend well beyond the diameter of the branch spread of the tree. They are shallow rooted and mostly exist within the top 30 cm (1 foot) of soil.
When a willow tree is cut down, the trauma to the tree's system sends a message to the roots that they need to expend their energy for regrowth. The remaining roots will send up multiple young shoots to try and take the place of the damaged tree. The ensuing growth is referred to as sucker growth, and a cut-down may send up dozens or even hundreds of these saplings after the tree has been felled.
Several options exist to control willow roots. The best option, if practical, is to have the stump ground out well below ground level. A tiller can be used if the tree was small and the root system not too vast. A broadleaved weed killer such as glyphosate is effective on the sucker growth, but expect to make repeat applications for total control.