Landscaping with weeping trees and shrubs should always be considered. Since they will attract the attention of the viewer, they should be planted in the background with other plants around them.
The picture of the full-grown weeping willow next to a pond always comes to mind. Just do not forget the many different evergreens and deciduous trees and shrubs you can use in your back or front yard. Cultivars range from a foot tall to over 60 feet with different textures, colors, and shapes.
For instance, a weeping river birch in the center of the backyard surrounded with yellow foliage barberries and perennials brings out the peach-colored bark since this tree does not cast a lot of shade. For willows try hostas which will take more shade.
If you want the tree to be the focal point, just create a rock garden around the trunk. Fewer weeds! A weeping tree always reminds me of water, so you can place a little garden fountain or pond next to it. Maybe add a bench underneath as a place to unwind after a hard day.
