It looks like oak trees don't need a lot to sustain growth.
Virtually every mature hardwood that you see is hollow or rotten in the center.
No growth at all takes place in the center of any tree, young or old. Both the xylem and phloem are contained in the outer sections, just beneath the bark. This is why girdled trees die - their food source is cut off by whatever is binding the outside of the tree.
More large trees than you care to imagine are closer to collapse than not. If they are completely upright/reasonable symmetrical, then there's probably no worries.
But if they are previously damaged, leaning (like along a riverbank - in the story), infested with insects (especially wood-destroying insects) or subjected to unusual winds or stress - well, there could be trouble.