Trees are one of the best things you can add to a slope to help prevent erosion. The roots will spread through the soil, holding it in place. One of the notable differences between planting on level grade versus planting on a slope is the depth of the root ball. There are a couple of things you should do to make sure your tree survives living on the slanted surface.
Mark the spot where you wish to plant your tree. This will be the center point of the hole you are going to dig for the root ball. Half of the hole will be in the hill and the other half you will extend straight out, on top of the surface of the slope. Mark a spot a little more than half the width of the root ball, straight up on the slope.
Dig straight down at the center point to the depth of your root ball, placing the soil on the slope below the hole, packing it down as you go. For instance, if you have a root ball that is 18 inches high, you will need to dig down 18 inches to keep the same soil line as the tree had in the nursery.
Remove the soil on the hill starting at the high point mark that you made earlier. Come straight down to where you have already dug, placing the removed dirt on the pile on the slope.
Remove the container the tree is housed in and set the tree in place. A smaller tree can simply be lifted to remove the container, while a larger tree might have to have its container cut away. Loosen any visible roots that may have started to spiral around and hold the tree straight up and down, using a level if necessary.
Fill in the soil around the roots, tamping it down with the heel of your boot as you go. By the time the whole root ball is covered, half if it should be in line with the slope and the other covered by soil on the outcrop you formed below the hole.
Form a swale on the slope just above the tree by removing an area of soil about 6 to 8 inches deep as wide as the tree to catch any rainwater. Otherwise, the rain will simply drain down the slope without absorbing enough moisture to sustain the tree's roots. Smooth it out and pile the removed dirt on the outcropping hump below the tree.
Mulch the tree heavily to hold the soil in place until the tree is established. Water the tree every day until you see signs of growth, unless you planted in the fall, in which case you should water daily for just about a week.