We consider the problem of justifying vertex weights of a tree under uncertain costs so that a prespecified vertex become optimal and the total cost should be optimal in the uncertainty scenario. We propose a model which delivers the information about the optimal cost which respect to each confidence level \(\alpha \in [0,1]\). To obtain this goal, we first define an uncertain variable with respect to the minimum cost in each confidence level. If all costs are independently linear distributed, we present the inverse distribution function of this uncertain variable in \(O(n^{2}\log n)\) time, where \(n\) is the number of vertices in the tree.