Special case of the $MST-$ Problem
Since the graph is complete, you can connect each vertex with any other vertex. Note that each vertex must be connected to the tree and hence has an incident edge in the tree.

Assuming all weights are positive, choose a vertex $v$ with the minimum weight and construct the tree from all incident edges with this vertex. The tree is hence a star graph with $v$ the center of the star.

Assuming the numbers can be negative (which is not specified in you question), then distinct two cases. the first one, all weights are negative. Hence, it turns into the case above, since  all products have to be positive.

The second case is where we have at least one negative weight and one positive weight. Let $u$ be a vertex with the maximum positive weight. Let $v$ be a vertex with negative weight with maximum absolute value among all negative weights. connect $u$ to all vertices with negative weights and $v$ to all vertices with positive weights. Be aware of not adding the edge between $u$ and $v$ twice.

Try to proof the correctness as an exercise. It should be clear using elementary math and an exchange argument.