Bootstrap trees and consistent S matrices

We analyze the tree structure arising from the recursive bootstrap equations, given the S-matrix of the lightest particle. When S11 contains only one singularity, among all possible bootstrap systems, the only ones which give rise to a consistent set of S-matrices coincide with those of breathers of sine-Gordon at the reduction point ξ=2π/(2n+1). We also present our investigation on bootstrap systems defined by a S11 with a higher number of singularities. The only consistent examples we found belong to the set of minimal S-matrices corresponding to Dynkin diagrams. (orig.)