The use of the (1/σ) expansion to calculate the thermodynamic properties of systems such as the Ising model or percolation whose diagrammatic expansion contains only diagrams with no free ends is reviewed. Here σ=z−1, where z is the coordination number of the lattice. For more general problems we formulate a self-consistency condition for a site potential h, so that diagrams with free ends are eliminated. Construction of h gives the leading order in (1/σ) solution and is exact for the Cayley tree. We obtain correction terms by using a bond renormalized interaction so that to order (1/σ)5 we need only consider two-site problems. Results are given for (1) Kc, the critical fugacity for animals, when either H, the fugacity for free ends, or Q, the density of free ends, is fixed, and (2) (t/Ec), where Ec is the mobility energy and t is the magnitude of the hopping matrix element whose sign is random. At d=8 our results appear to be accurate to within about 0.01% for both animals and localization. We also obtain an expansion for Q(Kc)/(zKc) whose divergence near spatial dimensionality d=4 supports the idea that the order-parameter exponent β for lattice animals passes through zero at d=4.