How about:
http://freemind.en.softonic.com/

TTFN
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I've seen mind maps before. It never occurred to me that they can produce the kind of tree in my original post. I realize that hierarchy is a very general concept used in many tools. I'm looking specifically for a tool that can produce the kind of fault-tree hierarchy shown in my original post. Is FreeMind able to do that? It must be a one-sided tree, collapsible at arbitrary nodes, and using logic AND/OR gates.

It wouldn't really be worthwhile for just one analysis, but if you're preparing many, or need many versions of similar analyses, then why not explore building the HTML and scripts to do it with a web browser. The scripts for collapsible option trees are fairly common, so read the code of a few webpages that use that structure and borrow any ideas that look good to you.
I don't know how to incorporate calculations into the web browser format, however there is some mathematic capability if you want to learn how to write PERL scripts that can receive and process web forms, and spit out the results. Fault trees are inherently arithmetic calculations, so I don't see an obstacle there. Just the learning curve is steep. Also, this way of doing analysis would be very hard to test or cross-check for accuracy, but it would be ideal if you need to expose your executive managers to an instant calculation of the ramifications of certain design changes.
Should you want to delve into it so far, of course...

STF

Unfortunately, automation development isn't something I'm able to pursue right now (though it would be fun). Otherwise, I'd consider doing this in Excel or Powerpoint using VBA. Which is why I was seeking a somewhat ready-to-use method. Not necessarily for the analysis, even just the graphical tree would be fine. It would sure beat manually finangling it, especially with a tool like Powerpoint. Unfortunately, using a better tool like Inkscape means replicating each tree diagram, once in inkscape and then in the powerpoint presentation.

By the way, I'm surprised that a tool doesn't exist for this specific form of the graphical tree. It seems so much more manageable than the typical digital-circuit style schematic.

There's this: http://www.openfta.com/Download.aspx

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Um...that really doesn't do what I described. It is generic FTA tool, but it is prone to crashing in a Windows 7 environment, especially with a large screen. No complaints are warranted because it is free, but I've been recommending that people go for a paid alternative for any serious FTA.