The roots of trees are important for providing the canopy with water, storing important minerals and sugars, and stability. Damaging them could harm or even kill the tree. Sometimes cutting roots in construction projects is inevitable, but minimizing the damage will minimize the potential for the tree to die or fall.  It is best to keep the patio out of the critical root zone where 85% of the mass of tree's roots are located. Birch trees typically have shallow root systems staying in the upper 2 feet of the soil depending on the soil depth. So, if you are cutting that deep, keep in mind that there aren't any tap roots or deeper roots that are still intact. It is also helpful to keep in mind that tree roots over 4 inches in diameter are likely to be structural roots; cutting these roots may impact the structural stability of the tree, creating the potential for the to fall over.  

So, how do you know where the critical root zone is?  The critical root zone radius is typically minimally at the drip line of the canopy of the tree. Another way to find the critical zone is to measure the diameter of the tree at 4 and a half feet off the ground and then convert the inches to feet and put the patio that many feet from the trunk of the tree. For example if the diameter of the tree is 6 inches then I wouldn't build the patio within 6 feet of the trunk. 

OSU Extension has a publication called Tree Protection on Construction and Development Sites (EM8994) that can help you even further with diagrams as well.  
Good luck with your new patio!