stringBuilder :: Tree -> [Char] -> [Char]
stringBuilder t s = snd (decode t s [])

-- Take the current tree, the current input string, and the current output
-- string, return the remainder of the inpupt string and the modified output
-- string
decode :: Tree -> [Char] -> [Char] -> ([Char], [Char])
decode (Leaf c)     i      o = (i, o ++ [c]) -- You only need one case here
decode (Branch a b) (i:is) o =
    case i of
        '0' -> decode a is o
        '1' -> decode b is o
        _   -> undefined
decode (Branch a b) []     o = ([], o)