What happens when running out of Cherry Trees in Tokaido: Crossroads?
You simply don't get one.  The rules for crossroads state :-


  When a Traveler stops at a Panorama (Sea, Mountain, or Rice Paddy)
  space, he must choose between:  • Taking a Panorama card of the
  corresponding type (basic game rules).  • Taking a Cherry Tree card
  (if the card is available)


So if a Cherry card is not available you just don't get one and must take the panorama. Of course remembering the basic game rules :-


  Each traveler can create only a single panorama of each type; a
  traveler who completes a panorama can no longer stop on the spaces
  corresponding to that type (Sea, Mountain, or Paddy).


To clarify If you have completed the panorama corresponding to that space and the cherry tree cards have all gone you may not choose to land on that space.