Want help with part (a). Other parts can be done accordingly.
According to the solution,
I understand how to find the limits of the sum, but why is there a factor of 2 with T(k) * T(n-k-1), according to my understanding it should not be there because T(k) is the count of left sub-trees and T(n-k-1) is the count of right sub-trees, so there are only T(k)*T(n-k-1) possibilities for each k, sum over the limits