First thing what came up in my mind was this forum-thread: How do I find the smallest possible bounding box for a solid? - #3 by RadovanG
It is about the smallest possible box around a solid.
Your question could be defined as the largest possible box inside a solid.
Second approach:
- Contour could be used to slice your tree in all directions: radial, horizontal, vertical to find the slice with the largest area.
Third approach: Assuming your largest possible cut will have the length of your tree, then you are sure that, there must be possible to draw 4 parallel lines (the corners of the cut) from one end to the other. You could use randomly placed points at both side and connect them. Cull all what cross the surface of the tree.
It might give you directions on how to search from there.
Nice project!. Regards, Eef