The question entails that I should choose two finite groups, then construct a 'biregular' tree, and show that the action of the free product of the two finite groups on the biregular tree will have a fundamental domain that consists of a single edge and two vertices. What I have so far is the two finite groups. The first group is $A = C_2$, and the second group is $B = D_4$. I know the group presentations of these groups. I understand that the free product of A and B is a group of symmetries of the biregular tree. I am lost on how to construct the biregular tree. If anyone can offer some suggestions or help it would be greatly appreciated. Thanks in advance.