Some birds settled on the branches of a tree. First, they sat one to a branch and there was one bird too many. Next they sat two to a branch and there was one branch too many. How many branches were there?

When the birds sat one on a branch, there was one extra bird.
When they sat Two to a branch one branch was extra.
Now, we go through the options;

Checking option (A);
If there were 3 branches, there would be 4 birds.
(This would leave one bird without branch as per the question.)
When 4 birds would sit Two to a branch there would be one branch free
(as per the question.).
Hence, the option (A) is correct.