Given a spatio-temporal network whose edge properties vary with time, a time-sub-interval minimum spanning tree (TSMST) is a collection of minimum spanning trees where each tree is associated with one or more time intervals; during these time intervals, the total cost of this spanning tree is the least among all spanning trees. The TSMST problem aims to identify a collection of distinct minimum spanning trees and their respective time-sub-intervals. This is an important problem in spatio-temporal application domains such as wireless sensor networks (e.g., energy-efficient routing). As the ranking of candidate spanning trees is non-stationary over a given time interval, computing TSMST is challenging. Existing methods such as dynamic graph algorithms and kinetic data structures assume separable edge weight functions. In contrast, we propose novel algorithms to find TSMST for large networks by accounting for both separable and non-separable piecewise linear edge weight functions. The algorithms are based on the ordering of edges in edge-order-intervals and intersection points of edge weight functions.