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Multiple independent spanning trees (ISTs) have applications to fault-tolerant and data broadcasting in interconnections. Thus the designs of multiple ISTs in several classes of networks have been widely investigated. There are two versions of the n ISTs conjecture. The vertex (edge) conjecture is that any n-connected (n-edge-connected) graph has n vertex-ISTs (edge-ISTs) rooted at an arbitrary vertex r. Note that the vertex conjecture implies the edge conjecture. Recently, Hsieh and Tu proposed an algorithm to construct -ISTs rooted at vertex 0 for the n-dimensional locally twisted cube (LTQn), which is a variant of the ndimensional hypercube (Qn). Since LTQn is not vertextransitive, Hsieh and Tu's result does not solve the edge conjecture for LTQn. In the paper, we confirm the vertex conjecture (and hence also the edge conjecture) for LTQn by proposing an algorithm to construct n vertex-ISTs rooted at any vertex. We also confirm the vertex (and also the edge) conjecture for Qn. To the best of our knowledge, our algorithm is the first algorithm that can construct n vertexISTs rooted at any vertex for both LTQn and Qn.