As transcoding caching is attracting an increasing amount of attention, it is important and necessary to find methods to distribute multiple versions of the same media object in the Internet. In this paper, we first present a mathematical model for the problem of optimally determining the locations in which to place multiple versions of the same media object in tree networks such that the specified objective is achieved. This problem is formulated as an optimization problem. Second, we propose a low-cost dynamic programming-based solution for solving this problem, by which the optimal locations are obtained. Finally, we evaluate our model on different performance metrics through extensive simulation experiments and compare the results of our model with those of existing models that consider transcoding caching either on a path or at individual nodes only.