Can one determinize finite automata over infinite trees?
This may be not explicitly spelled at many places in the literature, as deterministic tree automata are unable to capture specifications that are only expressible in branching-time logics.

Take for instance the property "There exists a node in the tree labeled with atomic proposition $x$" - this is equivalent to the CTL formula $\mathit{EF\,} x$. 
This one cannot be represented as a deterministic tree automaton, but is easy to express as a non-deterministic Büchi tree automaton.
The reason is that a tree automaton accepts a tree $\langle T, \tau \rangle$ if all branches in the run tree for $\langle T, \tau \rangle$ are accepting. With non-determinstic tree automata, one can route the obligation to see an $x$ to somewhere in the tree. With deterministic automata, this is just not possible and thus one can only express $\mathit{AF}\, x$ in this form.

Hence, non-deterministic tree automata cannot be determinized in general.