[FOM] branching quantifiers

[Thomas Forster]

I have two questions about them.  

(i) Does anybody know any useful facts relating expressive power 
of quantifiers with branching quantifiers to the *width* of the 
quantifier prefix? 

(ii)  Logic with branching quantifiers is equivalent in expressive 
to a logic allowing single higher-order existential quantifiers. As
far as i can see, this exploits the presence of `=' in an essential
way.  Am i right?  Does anyone know anything about the expressive
power of the branching-quantifier logic *without* equality?

      Thomas Forster