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The recently-introduced loop hierarchy method has been used to study a range of non-mean-field random operators, including a proof of the random band matrix delocalization conjecture. The loop hierarchy is a system of dynamical equations satisfied by a family of resolvent loops, and a key ingredient of this approach is the tree approximation, which is comprised of two main steps: (I) constructing “deterministic partners” with graph-theoretic tree representations to these (random) resolvent loops, and (II) showing that the resolvent loops remain stable—under suitable dynamical flows—near their partners. The analysis of the single-loop partners has already been successfully applied to random band matrices in [YY25; Dub+25b; Dub+25a] and to the block Anderson model for d= 1,2 in [TYY25]. In this thesis, we will push (I) further by giving a construction of multiple-loop and higher order partners. Calculating these partners is humanly intractable except in the simplest cases, and so a computer program was written to perform and organize these com- putations, which ultimately led to the results presented here. We believe that in future work, these results will be useful in the study of the distribution of resolvent loops, as opposed to just upper bounds, thereby extending the reach of the method to genuinely non-mean-field regimes.