Finding a minimum tree which is isomorphic to a subtree of $T_1$ but not to a subtree of $T_2$
You can put a tree into a cannonical form in $O(n)$ with leaf contractions. Read, Ronald C. (1972), "The coding of various kinds of unlabeled trees", Graph Theory and Computing, Academic Press, New York, pp. 153–182

When $T_{1}$ and $T_{2}$ are isomorphic you can solve it in $O(n)$.

Shamir's paper has $O({k^{1.5}\over log\ k}n)$ for testing if one tree is a subtree of the other, https://www.cs.bgu.ac.il/~dekelts/publications/subtree.pdf 

Not sure if that helps.