Here’s a path that could be taken by a chess king. All subpaths of length four describe a different pentomino:
This led from the grid of pentomino painting instructions that I posted previously. Consider a string of arrows for which the subsequences of length 4 include instructions for producing all 12 pentominoes. (This is somewhat analogous to a de Bruijn sequence.) For the case shown, the string is ←↑↖→→→→↓↓←↓←↘↗↓, although the graphic seems more illuminating than the arrow string here.
Instead of a path, we could have a tree of pentominoes:
Along with the constraints that each pentomino occurs exactly once, and no square is used more than once, I wanted to limit the number of branches per node. The root node having three branches might be considered a flaw, but this was the best I could do.