Tree-designs with balanced-type conditions

For a given graph G we say that a G-design is balanced if there exists a constant r such that for each point x the number of blocks containing x is equal to r. A G-design is degree-balanced if, for each degree d occurring in the graph G, there exists a constant r_d such that, for each point x, the number of blocks containing x as a vertex of degree d is equal to r_d.Let V_1, V_2, . . . , V_h be the vertex-orbits of G under its automorphism group. A G-design is said to be orbit-balanced (or strongly balanced) if for i = 1, 2, . . . , h there exists a constant R_i such that, for each point x the number of blocks of the G-design in which x occurs as an element in the orbit V_i is equal to R_i.If G is a tree with six vertices, we determine the values of v for which a balanced G-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.