The original tree has edges connecting nodes physically. The decomposition tree has edges connecting centroids based on removal order. These represent different relationships. A node's parent in the decomposition tree is the centroid that removed it. A node's children in the decomposition tree are centroids of components created by removing it.
These relationships do not match the original edges. These are different relationships. A node far away in the original tree might be close in the decomposition tree. This is useful for distance queries.