The volume of the undercut can be calculated with some calculus. If the tree were a perfect cylinder, a wedge-shaped piece is a solid called an "ungula"*. A tree is not a cylinder, though --- you can think of a tree as being considered as three parts; the lower part (the stump) as a neiloid (a solid formed by rotating a parabola around with its axis offset from the axis of the resulting solid), most of the tree (the part that becomes logs) as a straight taper, and the top of the tree as a paraboloid.
Equations exist that model a tree's shape pretty well, and I will show how to do this when I get around to formatting the equations so that they can be displayed here. Stay tuned!
*Deer, cows, etc., are ungulates --- their hooves look sort of like the wedge cut out of a tree for the undercut, right?