Well, the first rule of any DIY project is that whatever way you think
up that gets the job done is fair game.
If I were doing it, I would simply measure the width and height of the
opening and take a picture of it with a digital camera.
Then I would use software to enlarge the photo so that the opening in
the photo was the same size as I had measured.
Now, print the picture. Most printing companies have large format
printers that will allow you to print onto paper much larger than legal
Mark the opening on the large format print and cut it out. The cut out
is the correct size and shape of the opening you have to work with, and
can serve as a template by which to make the door.
You'd still have to do some carving into the tree or around the door to
improve the fit, but using a digital photograph would at least give you
a good first stab at it.
But, your point is well taken. Where there is no regular geometry to
guide you, fitting a door to such a random opening would be largely a
trial-and-error exercise whereby you make a crude door and measure how
far out it is, and then make another, and gradually zero in on the
correct size and shape.