Coded caching aims to minimize the network's peak-time communication load by leveraging the information pre-stored in the local caches at the users. The original single file retrieval setting by Maddah-Ali and Niesen has been recently extended to general Scalar Linear Function Retrieval (SLFR) by Wan et al., who proposed a linear scheme that surprisingly achieves the same optimal load (under the constraint of uncoded cache placement) as in single file retrieval. This paper's goal is to characterize the conditions under which a general SLFR linear scheme is optimal and gain practical insights into why the specific choices made by Wan et al. work. This paper shows that the optimal decoding coefficients are necessarily the product of two terms, one only involving the encoding coefficients and the other only the demands. In addition, the relationships among the encoding coefficients are shown to be captured by the cycles of certain graphs. Thus, a general linear scheme for SLFR can be found by solving a spanning tree problem.