Hello!
I just found this forum and think it's great. This was the first year that I dug a trench and overwintered my Hardy Chicago in the garden. When I removed the straw and earth I found that mice had eaten the bark around the bases of the branches. I thought I would be starting from scratch bot the branches have leafed out. They look completely girdled, and I was wondering if they will only get so big and die or can they get by like this. Should I cut off the branches and try to root them with some rooting hormone, ( is this posible?). Any help please.

Thanks Rog

Baitfish:
I find that when I girdle a tree it leafs out and grows the following spring.  However, if I have cut completely around the cambium layer the leaves cannot send food back to the roots and the tree dies in the following winter.

It would be my guess that your branches will react the same way.  If you have any sound branches you might cut off the girdled  branches and let the tree grow all the sound ones. 

I would NOT try to cut them off and root them after they had leafed out.   Instead, you might air layer the girdled branches and make new trees of each of them.  Once they have good roots each branch can be planted.

This is only opinion; I am sure there are people here with more "mouse experience" than I. 
Ox

Two choices:

1) Start air-layering the girdled branches. This is usually done when the branch is actively growing, with leaves.

2) Bridge-graft some bark across the girdled areas.

You can Google both techniques and find much useful info.

Bridge grafting is difficult and results in a weak joint.  Far better to air layer and start a new tree.
CS

Thanks for the good advice, air layering looks like a new adventure.

See "Air-Layering"