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We extend earlier results on conditioning of super-Brownian motion to generalbranching rules. We obtain representations of the conditioned process, both asan $h$-transform, and as an unconditioned superprocess with immigration along abranching tree. Unlike the finite-variance branching setting, these trees areno longer binary, and strictly positive mass can be created at branch points.This construction is singular in the case of stable branching. We analyze thissingularity first by approaching the stable branching function via analyticapproximations. In this context the singularity of the stable case can beattributed to blow up of the mass created at the first branch of the tree.Other ways of approaching the stable case yield a branching tree that isdifferent in law. To explain this anomaly we construct a family of martingaleswhose backbones have multiple limit laws.

Blowup and Conditionings of $\psi$-super Brownian Exit Measures