Closure under intersection of context free binary trees
If I correctly read your question there are two types of terminal symbols, disjoint sets for internal labels and leaf labels. In that way the structure of the tree is uniquely determined by the string.

Then your formalism is known as regular tree grammars. A production $A \to a BC$ seems to label a tree node by $a$ while attaching children $B$ and $C$. This formalism is closed under intersection. This can be proved precisely as for finite state automata or right-linear grammars. Simulate the two in parallel, as a product construction. With productions $A \to_1 a BC$ and $P\to_2 a QR$ we join them to $(A,P) \to a (B,Q)(C,R)$ and $A \to_1 a$ and $P\to_2 a$ are joined to $(A,P)\to a$.