When the arc weights are fuzzy numbers, the total weight of a spanning tree is also a fuzzy number. Since the number of spanning trees in a network is usually very large, how to select a spanning tree is an interesting problem. In this paper, based on the possibility theory, some rules and criteria for choosing a ranking function for this problem are proposed at first. Then, it is pointed out that if the ranking function is additive, then the problem can be transformed into the classical problem by defuzzifying the arc weights using the ranking function. In particular, weighted averages of the upper bound and the lower bound of the probabilistic mean value or the possibilistic mean value are suggested, in which the weight can be used to represent the characteristic of the decision maker. Finally, we consider the problem of finding the optimal spanning trees for all kinds of decision makers at the same time.