The Max-Sum algorithm is a solution method for the Distributed Constraint Optimization Problem (DCOP) which is a fundamental problem in multiagent cooperation. Particularly, we focus on the case of Max-Sum on a spanning tree, where the algorithm is an exact solution method. In this case, all agents simultaneously compute globally optimal objective values as erootf nodes of the tree that represents the problem. On the other hand, a tiebreak is generally necessary in order to determine a unique optimal solution among the agents. While top-down post-processing is a well-known solution, one can prefer to design the solver as a bottom-up computation that is simply integrated to pre-processing. To address this issue, we investigate a technique that employs a preference ordering based on spanning trees for the optimization algorithms. With this technique, top-down processing to choose a unique optimal solution can be embedded into bottom-up optimization via small weight values for the preference ordering. We also evaluate an integrated algorithm that maintains both tree structures and quasi-optimal solutions using the bottom-up approaches.