Suppose $\kappa$ is a weakly inaccessible cardinal with the tree property. What can we say about the height of $\kappa$? Is it a weakly-hyper-Mahlo of some sort? Does it enjoy some kind of indescribability property? Of course it is weakly compact in $L$, but I am interested in what height properties we can say it has in a universe where $\kappa$ is not strongly inaccessible.
In his paper Boolean extensions which efface the Mahlo property William Boos proves the following consistency result:
Theorem. Assume GCH holds and $\kappa$ is weakly compact. Then there exists a cardinal preserving generic extension of the universe in which $\kappa$ is the least weakly Mahlo cardinal and the tree property holds at $\kappa$.
The proof of the theorem is very similar to that of Mitchell, the main difference is that at Mahlo cardinals below $\kappa$ he adds a club of singular cardinals into that cardinal.
The following is asked in the paper and is still open:
Question. Is it consistent that the tree property hold at the least weakly inaccessible cardinal?