Question: If one has built a kasher sukkah, with its own completed skhakh but there is a tree branch that leans against the skhakh, is it still kasher?

Answer: 
There are many details found in the Shulchan Aruch siman 626. 

If the branch only provides a minority of the shade etc. In one situation having the branch mixed in with the kosher schach is actually beneficial. 

But according to one opinion in the Shulchan Aruch, one should never under any circumstances have any branch above the schach.

If you rule with Ramma or straight Shulchan Aruch will come into play here, so again, see the details there.
Since the living branch covers only a small part of the succa, you can just not sit under the branch. Most succot (these days) are so large that a small area of invalid roof does not invalidate the whole succa. (See Succah 13b, I think.)