I want to thank everyone who commented on this thread. In retrospect the comments were very correct, and helped me to be a better informed Dad throughout this process. My daughter of course took the lead in this process and I did little more than comment and cheer, but it was very helpful to be able to understand the process well while commenting and cheering.
As a parent who comments on this web site, I like to see how things turn out, particularly when they turn out well. I thought I would give the good news related to this case.
My daughter tends to be very thoughtful and think things through carefully before acting. In this case she did think things through carefully and determine that going for a PhD is the right path for her. I do not know whether she ever decided which of the programs that she applied to was her first choice, but she was leaning a bit one way and did get accepted to a very good PhD program that was most likely her top choice. This appears to be another case of a very good university and a very strong student each looking for and finding a good fit.
We are very excited.
Thanks to everyone for your help.