A method for obtaining a 1/σ expansion for certain statistical models is presented, where σ+1 is the coordination number of the lattice. The method depends on being able to generate exact recursion relations for the Cayley tree. By perturbing the recursion relation to take account of the dominant loops in a hypercubic lattice for large σ, we obtain corrections of order σ−2 to the recursion relations. For the tight-binding model on random bond clusters (the quantum-percolation problem) we obtain corrections to this order for the critical concentration p* at which the transition between localized and extended zero-energy eigenfunctions takes place. It is believed that this concentration coincides with the transition when all energies are considered. In addition, we display the relation for the Cayley tree between quantum-percolation and lattice animals (or dilute branched polymers). We show that this relation manifests itself in the appearance of singularities in the quantum-percolation problem at negative concentration which correspond to the physical transition at positive fugacity in the statistics of lattice animals. Corrections of order σ−2 to the location of this unphysical singularity in the quantum-percolation problem are also obtained.