  Given a finite group G and a natural number n, we study the structure of the
complex of nested sets of the associated Dowling lattice Q(G) and of its
subposet of the G-symmetric partitions Q_G which was recently introduced by
Hultman together with the complex of G-symmetric phylogenetic trees T_G.
Hultman shows that T_G and Q_G are homotopy equivalent and Cohen-Macaulay, and
determines the rank of their top homology. An application of the theory of
building sets and nested set complexes by Feichtner and Kozlov shows that in
fact T_G is subdivided by the order complex of Q_G. We introduce the complex of
Dowling trees T(G) and prove that it is subdivided by the order complex of Q(G)
and contains T_G as a subcomplex. We show that T(G) is obtained from T_G by
successive coning over certain subcomplexes. We explicitly and independently
calculate how many homology spheres are added in passing from T_G to T(G).
