There are many different species of trees with shallow root systems, including cottonwood, beech, birch, poplar and several types of maple. Many shallow root systems are considered invasive, as they grow quite quickly, such as willows and spruces.
In addition to spruce trees, most other evergreens also have very shallow roots systems, as they have adapted to live in very shallow soil that is often mostly devoid of nutrients. The soil is generally very acidic due to the large amounts of decaying pine needles, meaning almost no other life can grow near an evergreen.
Sugar maple, Norway maple and silver maple all have extremely shallow roots, especially silver maples, which often have large, exposed roots well above the ground level. This makes it difficult to nearly impossible for grass or other plants to live underneath a silver maple tree.
Cottonwoods, aspens and poplars don't just have shallow roots, but also huge root systems that spread out over a great distance. The roots of these trees are incredibly aggressive and can cause extensive damage to anything they encounter, which is why they should never be planted near homes or other buildings. There are also a number of hybrid poplar species that have been crossbred to grow incredibly quickly, but can wreak havoc on their surroundings during their short lifespan.