Minimum cost spanning tree problems try to connect agents efficiently to a source when agents are located at different points in space and the cost of using an edge is fixed. We examine solution concepts to divide the common cost of connection among users, and revisit the dispute between two of the most familiar solutions: the Kar and folk solutions, both based on the familiar Shapley value. We characterize of family of solutions made up of solutions that are a weighted sum of the Kar and folk solutions, distinguished by a single parameter that turns out to be how we share cost in a simple two-agent problem. This family is characterized using a new property called Weak Problem Separation that allows, under conditions, to divide the problem in two: the connection of an agent to the source and the connection of agents to each other. New characterizations of the Kar and folk solutions are then offered. In addition, a new rule is proposed and characterized.