Mulberry tree issue

I have this poor fruitless mulberry tree in my front yard that I am
trying to care for.
We just moved recently to a new home.
I noticed two large branches were pressing against each other.
>From reading about trees I think I need to cut one of the branches
off.
First of all I want to verify this and second of all if I do so is
there anything I need to watch out for and need to know?
Thank you everyone,
Anita
Here is are some pics: 1.http://s87.photobucket.com/albums/k125/
soive2000/?action=view&current=Fruitlessmulberry4.jpg

2.http://s87.photobucket.com/albums/k125/soive2000/?
action=view&current=Fruitlessmulberry3.jpg

3.
http://s87.photobucket.com/albums/k125/soive2000/?action=view&current=Fruitlessmulberry2.jpg

4.
http://s87.photobucket.com/albums/k125/soive2000/?action=view&current=Fruitlessmulberry1.jpg