# Why are regular tree languages closed under intersection, but deterministic context free languages are not closed under intersection?

I am looking for intuition here. Essentially, I understand that the set of parse trees from a context free grammar forms a regular tree language. I also understand that regular tree languages are closed under union, intersection and complement. Now, why are deterministic context free languages not closed under intersection? (the intersection of DCFL need not even be in CFL) but intersection of regular tree languages are closed under intersection? (I get why DCFL union is not in DCFL -- because there is ambiguity).

For the intersection of two context-free languages we need two trees, one for each grammar and we only need the two frontiers to be the same. Hence we can find languages like $$\{a^nb^nc^n \mid n\ge 1\}$$ which is the intersection of two context-free grammars. The first one has a tree structure that verifies that the $$a$$'s and $$b$$'s are equal, while the second grammar checks the $$b$$'s and $$c$$'s with another tree structure. Also we need to 'guess' the two trees as we only 'see' their frontiers.