There was recently a funny case in Berkeley where the university or city wanted to cut a few trees, reasonably new trees, and people started to live on them, the usual nude protests, etc. Was there any importance in preserving that particular tree or trees besides the boredom and need for exhibition of some people? Ok, it's a beautiful tree which I like so I'm glad it was saved, but it was not in any sense historic or even important to save besides its prettiness.

This is part of the problem of the preservationists - they need to try to make a clear manifesto on what must, what should, what could, and what shouldn't be preserved. We try to preserve all information possible, and our subconscious can maintain many details we supposedly forget, but this is impossible in cities if we want to allow them to grow. One has to be allowed to forget.

Historic preservation in America has a clear set of guidelines through the listing criteria of the National Register of Historic Places.  To be eligible for listing in the National Register, which is kept by the National Park Service, a site has to be at least fifty years old - as long as it's fifty years old, it's technically "historic".  Beyond this, significance is determined by the site's association with historic people or events - not necessarily by its architectural merits.  A great, recent example is Natl Historic Landmark site in Delano, CA where Cesar Chavez worked to organize the United Farm Workers of America - it's basically a cinderblock bunker, with 40 acres of land attached.
But I'm guessing that "the preservationists" referred to above are not the ones working to preserve unimpeachably significant sites, like the Chavez site.  You're probably talking about people who want to keep things that aren't beautiful in any widely recognized way and aren't associated with any particular person or event - i.e., most of the built environment.  Personally, as a preservation professional, I think it's not hard to make a case for preserving pre-war architecture of any kind, given that there's so little of it left in this country and so little built history in America to begin with.  But with post-war construction, it's more complicated.  The fifty-year rule made a lot more sense when the National Register was created in the 1960's, since modern architecture and materials, which are often less durable and more controversial in style, hadn't entered the mix yet.  

It's complicated not only by the challenges of modern materials conservation, but by modern architectural styles, many of which don't necessarily resonate with today's public.  This will probably change - check back in fifty years - but it's an issue now.  And beyond aesthetics, there's a conscience problem, too, with the architecture associated with urban renewal, which often replaced dense, pre-war residential or commercial buildings with imposing civic or institutional structures.  Today, for people who know about the history of these places, these sites and buildings are like metaphors for a top-down decision-making process that failed to recognize the the integrity of existing communities.  But they also demonstrate a civic commitment to big ideas, a sense of idealism, and a particular notion of progress, as well as some interesting/challenging architecture.  The situation with Boston City Hall is a great example of all of these issues.    
So to answer your question, yes, there are criteria.  Yes, it's complicated.  And maybe I didn't directly speak to this last part, but yes, it's important to save things, because how else can we learn?

Thanks Anne, it's enlightening what you say. But let's even just look at the point of historical significance. If a few years ago it was a simple attribution (Picasso lived there etc.), today everything is historical. A house of a participant of Big Brother or another TV show could be called "historical" by some. That was my point with the link to the ubiquitous Facebook photos, that today every moment, and person, is historical. The fact that today everything is historical makes nothing historical.

You also mention that "I think it's not hard to make a case for preserving pre-war architecture of any kind." Let's say you had to choose and couldn't preserve all of them - how would you go about making that choice? There are 1000 pre-war buildings without historical significance and you get to preserve 100. If it's by aesthetical preference that could be a matter of taste (unless it's by a famous architect, which then makes it historical). For example, is uniqueness of the architecture a plus or a minus in this case?

I should say that I certainly think preservation is very important, even of what we would consider ugly buildings, It was just not clear to me how do people go about choosing what to save. I can say for example that attempts I've seen to save several "historical" cinemas have failed, only to be replaced by clothes shops.
I think understanding this question about buildings will be revealing also about what memories do we choose to keep (e.g. post), as John alludes to, and as you do with your very interesting urban renewal example.