The idea of embedding the Wasserstein distance in a tree-metric is an interesting idea accompanied with a thorough study and showing good empirical results. The contribution could make the Wasserstein community more aware of tree metric techniques. However, the presentation of the state of the art and the position of the contribution must updated in a correct way and we urge the authors to do it in an appropriate manner. In particular, the fact that (i) any metric can be embedded into a random ensemble of trees with small distortion, (ii) a good way to do it tree-of-hypercubes, (iii) this is useful for approximating OT and for nearest neighbor search in the OT metric, (iv) OT on trees is an l_1-metric are clearly contributions related to ref [37]. The appropriate reference for algorithm 1 must also be clearly mentioned. Authors should take into account the remarks and suggestions made by reviewer 3 and check appropriately the literature. The position of the contribution with respect to the existing literature must be highlighted clearly. This is beyond a classic camera-ready copy work, but this is essential from a scientific point of view. Many thanks for your consideration.