Question on "Searchable Symmetric Encryption: Imroved Definitions and Efficient Constructions" paper

Hi! I'm reading a paper from Curtmola et. al. titled "Searchable
Symmetric Encryption: Imroved Definitions and Efficient Constructions"
that describes techniques for searches on encrypted documents and got
stuck at one place.

Namely, I can't understand why in SSE-2 construction in order to
perform a search for a word a client should search for 'max' labels in
the word's family, where 'max' is the size of the largest plaintext
document. It seems to me that the client should perform search for
'number of the documents in the collection' labels, since the searched
word may occur in any document.

If there is someone familiar with this work, could she point out what
I'm missing?

Sergei

.