Summary: Distributed optimization has garnered significant attention in past decade, yet existing algorithms mainly rely on Laplacian matrix information for parameter settings, limiting their adaptability and applicability. To design the fully distributed algorithm, this paper uses an adaptive weight framework based on directed spanning trees (DST), which not only solves the consensus optimization problem but also can be extended to solve the resource allocation problem. The innovative integration of Nabla fractional calculus further improves performance, enabling efficient discrete-time distributed optimization. Moreover, The proposed algorithms optimality and convergence properties have been rigorously analyzed, which demonstrates that they can converge to the optimal solution of the problem under consideration. Finally, numerical simulations are conducted to validate the algorithm's feasibility and superiority.