The spectral dimension of generic trees

We define generic ensembles of infinite trees. These are limits as N -> infinity of ensembles of finite trees of fixed size N, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension d(h) =2. Our main result is that the spectral dimension of the ensemble average is d(s) =4/3, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is 1/3.