N-qudit SLOCC equivalent W states are determined by their bipartite reduced density matrices with tree form

It has been proved that N-qudit (i.e., d-level subsystems) generalized W states are determined by their bipartite reduced density matrices. In this paper, we prove that only $(N-1)$ of the bipartite reduced density matrices are sufficient. Furthermore, we find that N-qudit W states preserve their determinability under stochastic local operation and classical communication (SLOCC). That is, all multipartite pure states that are SLOCC equivalent to N-qudit W states can be uniquely determined (among pure, mixed states) by their $(N-1)$ of the bipartite reduced density matrices, if the $(N-1)$ pairs of qudits constitute a tree graph on N vertices, where each pair of qudits represents an edge.