I don't think the question is incomplete they are saying connected graph without cycle which means its a tree which is a bipartite graph! so no need to mention it explicitly.
Every tree we can divide into two groups such that no two vertices in a group share an edge. Suppose we have done that and we have two groups-A and B.
Now, from 5 colors we can select 2 different colors in 5C2 ways. Suppose we selected RED and BLUE.
we have two ways to assign R and B to these 2 groups. A-Red B-blue and A-Blue B-Red.
Total 5C2*2=20 ways.