A Birdsmouth consists of two cuts, the horizontal seat cut and the vertical heel cut. Based on what you enter for either cut, you can get the depth of the second cut.
If you are cutting a birdsmouth in your rafter, your local building code will limit the depth of the notch into the rafter. It can be rafter size divided by 3, 4 or whatever the local requirement is.
The limiting factor can be the minimum size of the seat cut that sits on the wall. The International Residential Code stipulates 1 1/2 inches of minimum bearing on the top plate, and here in our location, the rafter requires a minimum of 3 inches bearing on the top plate.