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I am assuming you mean a binary rooted tree. I don't know your definition of complete. The group is the infinite permutational wreath product of symmetric groups of degree 2. Good references are Automata, dynamical systems and infinite groups by R. Grigorchuk, V.V.Nekrashevich, V.I.Sushchanskii, Proc. Steklov Inst. Math. v.231 (2000), 134-214 and Cyclic renormalization and automorphism groups of rooted trees by Bass, Otero-Espinar, and Rockmore.