First of all, we use Master's theorem, when the recurrence are of form T(n) = a T(n/b) + f(n)
Tree method is much helpful when the problem is divided into two uneven fractions like
T(n) = T(n/2) + T(n/3) + c
But to solve square root recurrence better you go with substitution.
Now, let me tell you, what is problem with you approach,
S(m) = T(2m)/2m
S(m/2) = T(2m/2)/2m/2
this way T(2m/2) will become 2m/2 S(m/2) not simply S(m/2).
So your recurrence will become
S(m)=2*2m/2*S(m/2)+1
now to solve this recurrence, you cannot use master's theorem, so go for substitution method only.
For more help refer this link : http://cs.stackexchange.com/questions/6410/solving-a-recurrence-relation-with-%E2%88%9An-as-parameter