Tree-designs with balanced-type conditions

If G is a tree with six vertices, we determine the values of v for which a balancedG-design with v points exists, the values of v for which a degree-balanced G-design with v points exists, and the values of v for which an orbit-balanced G-design with v points exists. We also consider the existence problem for G-designs which are not balanced, which are balanced but not degree-balanced, and which are degree-balanced but not orbit-balanced.