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For all $n\\in\\mathbb{N}\\cup\\{\\infty\\}$, the Galois groups $G_n(\\beta)=\\text{Gal}(K(f^{-n}(\\beta))/K)$ embed into $\\text{Aut}(T_n)$, the automorphism group of the $d$-ary rooted tree of level $n$. A major problem in arithmetic dynamics is the arboreal finite index problem: determining when $[\\text{Aut}(T_\\infty):G_\\infty]<\\infty$. 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