Very good question. Depth of field is decided by a number of things. These are primarily, aperture, effective focal length, focus distance (distance from lens to subject), and the subject's distance from the background.
Simply put, to achieve a shallow depth of field you should get your lens objective as close to the subject as you can, and use the largest aperture and longest focal length possible. As you rightly said, you should also attempt to ensure that your subject is as far away from the background as possible.
You're the first person to ask about this, and I may well do a post on this subject in the future. It does get quite technical however when you start talking about things like, the effective field of view and circle of confusion. If I get some time next week though, I may well do a blog in relation to this.
I'm glad you enjoyed the images, and in each instance I was fairly close to the birds, and the backgrounds were sufficiently far enough away so as to be completely obscured. Anyway, thanks again for your question, cheers for now.
Thanks. I always enjoy hearing how images are made. Gives me ideas for other things to try and keeps me from getting stuck in a rut. (I also really appreciate that you post your cameras settings!) I have not yet done much wildlife photography, but just recently got a longer lens for my Fuji X cameras and took it to the zoo to try it out. Very excited about the possibilities!