the <Kennedy, Kennedys' or Kennedy's> family tree

My daughter had an English exam. She was a bit disappointed about the result and asked me whether this and that was really a mistake. I said yep, sure is - but wait, this seems correct to me:
She had written: ...the Kennedy family tree...
The teacher corrected: ...the Kennedys' family tree...

Somehow it seemed to me that the name and family tree melted together to an expression. The next moment I had my doubts. I guessed the plural genitive case to be correct after all. Only to settle for: either way.

But how to prove. So I entered the terms in Google with some amazing results

„the kennedy family tree“ – 665
„the kennedy’s family tree“ – 3

„kennedy family tree“ – 10.600
„kennedy’s family tree“ – 241

The numbers were definitely skewed away from genitive singular. I communicated the results to the teacher but, oh how embarrassing, genitive singular would never had come to my mind. Nonetheless it seems that genitive singular is still more popular than genitive plural:

"the kennedys' family tree" - 2 occurrences
"kennedys' family tree" - 9 occurrences

Now, I'm not sure whether this is due to problems with the apostrophe. But this surprised me.

What is correct, and foremost, what is the sophisticated explanation?

Try here

A great section on possessives'/possessive's/possessives :lol::roll:

or on this site http://www.usingenglish.com/weblog/archives/000067.html

Bringing this to a close let me share with you an eMail I received. It gave
me the theoretical background I was looking for. I now know that the non-possessive form is called classification.
(note: to simplify I took "the Kennedy children" vs. "the Kennedys' children")

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Your daughter's teacher humbles me--I will need to increase the rigor
of my own teaching!

The use of a classification (Kennedy) rather than the possessive
(Jameses'--child of both Mr/Mrs James--or, in this case, Kennedys') is
stylistic rather than grammatical--the New York Times, for example,
would prefer the latter, while often television newsbroadcasts opt for the
former, hoping for clarity. The instructor who chose this question may
have been testing for whether students would pluralize Kennedy and use
the apostrophe afterwards. Using classification instead was a clever and
knowledgeable alternative--I hope her instructor will agree!
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