A little bit for the bushes, not much for the trees. Since there is a horizon line the scene should have some perspective. The viewer should look down just a bit on short objects like the bushes. Tall trees will stretch across the horizon line so you can get away with less perspective. For the trees, make the base curved instead of flat and you're probably fine.
To be honest the degree of perspective tends to vary quite a bit from cartoon to cartoon. There are also perspective tricks to set up the backgrounds with multiple vanishing points across large horizontal layouts, but I'm not sure you need to use them.
One important trick is to hide the explicit breaks in the depth areas. Those lines that I drew? They should not be obvious to the viewer. Notice that the ground around the tree. The shadow/grass spills into the character area to hide the face that there is a hard break in the depth. Same thing on the right hand edge where the ground goes from tan to greenish. This makes the clearing a sort of round, space like a room, even though the character will never step in there depthwise.
The (red) line between mid ground and background is explicit on the right to keep them separate, but on the left the line is broken up by the large rocks and extra trees. Other cartoons place the objects at slanted angles relative to the horizontal axis, but the actual boundaries for the characters and objects remain a fixed horizontal band to make it easier on the character animators.


from Hiawatha's Rabbit Hunt, 1941. This is a Bugs Bunny cartoon.
A little perspective on the bushes, not much on the trees. A wide empty space at the bottom for the characters to walk around in. Note how all of the boundaries are curved or angled rather than horizontal. This is illusion for the most part.
The canoe is technically at the edge of the middle area, even though the lake above it is background. You'd have to do a custom animation for a character to get in or out of the canoe.
Browse the blog for more examples (it's not mine, but I think it's rather nifty).