Q.killing grass under a mature tree
We need to kill grass that is under a mature liquidambar tree. The tree is about 18″ diameter trunk and 35′ tall. The plot is about 4′ wide by 25′ long (parkway between sidewalk and street). The roots of the tree are mostly below the soil, but have lifted the area where the grass is like a mound, so if we dig down, I am sure we hit the surface roots pretty quickly. We want to use Round-up, because it is effective, but don’t want to kill the tree.

I personally would recommend removing as much of the sod using a square shovel. Then using one of the methods in the article below to kill the remaining grass naturally. This would be the safest way to remove the grass and keeping the tree and it's roots safe.

Use an agricultural chemical called Fusilade Forte*, I think about 100-125 mls to 10 ltres Of water around a mature tree. This can also be used for spraying grass in gardens at 15-20 m ls to 5 litres of water as it will not affect most garden plants.
* Fusilade is selective so quite expensive at 80-90 dollars for 1 litre and is available from Norco, Land care, National Farmers warehouse and the like.
Hope this is of some help,