Does all derivation trees generated by context free grammar in cnf form can be recognized by buttom up tree automata?

G is a context-free grammar in Chomsky normal form.

we define L(G) to the set of all derivation trees that formed by G.

Is it possible to create a non-deterministic bottom-up tree automaton that will accept L(G) exactly? if so, how to construct such automaton?

I think it’s true, so I’m trying to construct the automaton but having hard time to define specifically the transition function.

hope to get help.