In this paper we find trees with minimal and maximal Merrifield-Simmons index over the set $\Omega \left( n,2\right) $ of all trees with $n$ vertices and $2$ branching vertices, and also over the subset $\Omega ^{t}\left( n,2\right) $ of all trees in $\Omega \left( n,2\right)$ such that the branching vertices are connected by the path $P_t$.