Exotic trees.
We discuss the scaling properties of free branched polymers. The scaling behavior of the model is classified by the Hausdorff dimensions for the internal geometry, d(L) and d(H), and for the external one, D(L) and D(H). The dimensions d(H) and D(H) characterize the behavior for long distances, while d(L) and D(L) for short distances. We show that the internal Hausdorff dimension is d(L)=2 for generic and scale-free trees, contrary to d(H), which is known to be equal to 2 for generic trees and to vary between 2 and infinity for scale-free trees. We show that the external Hausdorff dimension D(H) is directly related to the internal one as D(H)=alphad(H), where alpha is the stability index of the embedding weights for the nearest-vertex interactions. The index is alpha=2 for weights from the Gaussian domain of attraction and 0<alpha<2 for those from the Lévy domain of attraction. If the dimension D of the target space is larger than D(H), one finds D(L)=D(H), or otherwise D(L)=D. The latter result means that the fractal structure cannot develop in a target space that has too low dimension.