```haskell
invert :: Tree a -> Tree a
invert (Leaf x) = Leaf x
invert t@(Node _ _) = Node (invert (pick False t)) (invert (pick True t))
  where
    -- pick b t = subtree consisting of exactly the leaves whose *last* step is b,
    -- with that last step dropped (so the depth decreases by 1).
    pick :: Bool -> Tree a -> Tree a
    pick _ (Leaf x) = Leaf x  -- not expected for perfect trees, but keeps the function total
    pick b (Node l r) =
      case (l, r) of
        (Leaf x, Leaf y) -> if b then Leaf y else Leaf x
        _                -> Node (pick b l) (pick b r)
```

(If you want the provided `main` to compile, make sure `Tree` derives `Show`, e.g. `data Tree a = ... deriving (Show, Eq)`.)