A set of spanning trees in a graph is said to be independent (ISTs for short) if all the trees are rooted at the same node $$r$$r and for any other node $$v(\ne r)$$v(?r), the paths from $$v$$v to $$r$$r in any two trees are node-disjoint except the two end nodes $$v$$v and $$r$$r. It was conjectured that for any $$n$$n-connected graph there exist $$n$$n ISTs rooted at an arbitrary node. Let $$N=2^n$$N=2n be the number of nodes in the $$n$$n-dimensional Mobius cube $$MQ_n$$MQn. Recently, for constructing $$n$$n ISTs rooted at an arbitrary node of $$MQ_n$$MQn, Cheng et al. (Comput J 56(11):1347---1362, 2013) and (J Supercomput 65(3):1279---1301, 2013), respectively, proposed a sequential algorithm to run in $${\mathcal O}(N\log N)$$O(NlogN) time and a parallel algorithm that takes $${\mathcal O}(N)$$O(N) time using $$\log N$$logN processors. However, the former algorithm is executed in a recursive fashion and thus is hard to be parallelized. Although the latter algorithm can simultaneously construct $$n$$n ISTs, it is not fully parallelized for the construction of each spanning tree. In this paper, we present a non-recursive and fully parallelized approach to construct $$n$$n ISTs rooted at an arbitrary node of $$MQ_n$$MQn in $${\mathcal O}(\log N)$$O(logN) time using $$N$$N nodes of $$MQ_n$$MQn as processors. In particular, we derive useful properties from the description of paths in ISTs, which make the proof of independency to become easier than ever before.