Company SuperRich is considering the possibility of capacity expansion. The company has three options. The first is to build a small facility that it could build at a cost of $$5$ million. If demand for new products is low, the company expects to receive $$7$ million in revenue with the small facility; if demand is high, it expects $$12$ million in revenues using the small facility. The second option is to build a large factory at a cost of $$10$ million. If demand is low, the company would expect $$9$ million in revenues with the large plant; if demand is high, the revenue is expected to be $$20$ million. In either case, the chance of high demand is $0\mathrm{.}4$ and low demand is $0\mathrm{.}6.$ The third option is not to construct any new facility, which will lead to $$5$ million revenue regardless of the demand condition.

a$.$ Construct a decision tree and conduct the analysis to help company SuperRich to make the best decision.