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We study Thurston equivalence classes of quadratic post-critically finitebranched coverings. For these maps, we introduce and study invariant spanningtrees. We give a computational procedure for searching for invariant spanningtrees. This procedure uses bisets over the fundamental group of a puncturedsphere. We also introduce a new combinatorial invariant of Thurston classes -the ivy graph.

Invariant spanning trees for quadratic rational maps