In the puzzle game Pipes, a space-filling tree of pipes (which may have either one, two, or three neighboring connections each, but not four) is scrambled by a series of tile rotations, and the goal is to reconstruct the solution. To do so, you may rotate any tile, including the "source" tile (marked with a red circle), any multiple of 90°. (Note: "space-filling tree" means every tile/square must have a pipe part on it, and no loops are allowed)
Main puzzle statement: Is it possible for a Pipes puzzle to have multiple solutions (equivalently: is it possible to transform one tree into another via rotations)? What might be the smallest $N$ for which this can happen on an $N \times N$ grid?
Easier analogue: Pipes Lite is a fictional alternative in which pipes may have only one or two neighboring connections each. Is it possible for a Pipes Lite puzzle to have multiple solutions?
More difficult analogue: Is it possible for a Pipes puzzle to have no possible tile orientation deductions from the outset? That is to say, any individual tile has at least two orientations belonging to separate solutions?