In conditional probability we use P(A|B) = P(A and B)/P(B). So basically you are finding P(One is a Boy AND Other is Boy)/P(One is a boy).
Now P(One is a Boy) = P(First is a boy) + P(Second is a Boy) - P(Both Are boys) = 0.5 + 0.5 - 0.5*0.5 = 0.75 which matches the tree diagram.
P(One is a Boy AND Other is a Boy) = 0.25 since both are independent of each other and the only possibility is BB.
So your answer is 0.25/0.75 = 1/3.
If you don't follow what I've done, then please say so to stop any confusion.