I'm trying to determine why web sites use sub domains. I have read
that its purpose is branding.  From a search engine optimization
standpoint, is there any evidence that the use of sub domains has any
significant advantage.

Why is:

answers.google.com

preferable to

www.google.com/answers

Hi, 

Typically subdomains are setup for traffic flow and load balancing on
large websites, not really for marketing reasons. A subdomain can be
set to its own IP address, where as a directory can not, unless you do
some really good hacking in the server area. So if you have a huge
website, that has a great deal of traffic, such as Google, and you
want to add a service to it, and keep the main domain name, then you
might look at using a subdomain, thus placing the new service on
another server (or servers, in this case). This keeps things running
smoothly and its easier to maintain as well.

Some writeups on this

DomainPeople.com
http://www.domainpeople.com/supportfaqs-dns.cfm

The Static Load Balancing Problem
http://www.dl.ac.uk/TCSC/Staff/Hu_Y_F/PROJECT/pdcp_siam/node3.html

One thing to keep in mind with the use of subdomains this that most
search engines treat them as separate sites, meaning that your Page
Rank and listing status may be affected greatly if you break your
website up in this manner. So if you don't have a large site with a
great deal of current traffic, I would advise against the use of this
schema, and go with directories instead.

webadept-ga

Great answer.  Thanks again.  Boy, you're fast.

Sadly, I have to say this answer isn't completely accurate. 

All the technical information is true but Sub domains are indeed use
also as a branding strategy. Yahoo.com is probably one of the most
know. (Ex: news.yahoo.com, finance.yahoo.com) Replacing the ‘www’ by
something else may help consumer recall your domain name more easily.

Montreal base mall Plaza-StHubert present their website like ‘an
address like no other’, they also run print ad for their website with
this tag line. The website use 4 w instead of 3.
(wwww.plaza-st-hubert.com)

We also use it to track down advertising effort. For example, print ad
may have a different subdomain so the company can have picture of
performance for their print strategy. We try to use subdomain because
it’s more easily recall then asking user to log in a subfolder (ex:
www.test.com/printad/ instead of printad.test.com)

Sorry for my English. Have a nice day !