I agree with Michael, cut-out dodgers often aren't worth the bother.
But, if you are going to do it:
You should dodge 'inside' the tree. The dodger needs to be cut down from the tree so that the wiggled dodger doesn't go outside of the tree boundaries - think 'thick snake'. A dark inside edge to the tree will look far more natural than a halo. It is, of course, impossible to make an exact dodge with a cardboard cut-out.
If you really need an exact dodge then the best method is to use photoshop.
In the old days this problem was solved by making two prints at very large size - one for the tree and one for the background. The tree is then cut out from it's print and glued onto the background print. Retouch fluid and gesso are used to hide the cut & paste line. A copy negative is then made for producing the final print. See "Lootens on Enlarging and Print Quality". (Gee, it does sound like a photoshop procedure, doesn't it?)