Wiener index of subdivisions of a tree

The Wiener index $W(T)$ of a tree $T$ is defined as the sum of distances between all vertices of $T$. The edge $k$-subdivision $T_e$ is constructed from a tree $T$ by replacing its edge $e$ with the path on $k+2$ vertices. Edge $k$-subdivisions of each of edges $e_1, e_2, \dots, e_{n-1}$ of a tree with $n$ vertices generate a family containing $n-1$ trees. A relation between quantities $W(T_{e_1}) + W(T_{e_2}) + \cdots + W(T_{e_{n-1}})$ and $W(T)$ is established.