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][1] 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.


  [1]: http://www.math.tamu.edu/~grigorch/publications/PSIM128.PS