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 sine-Gordon breathers at the reduction point zeta = 2-pi/(2n + 1). We also present our investigation of bootstrap systems defined by an 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.
