A 2- to 4-inch layer of organic mulch around a weeping cherry serves many purposes. Weed reduction, soil-moisture conservation, reduction of soil compaction from foot traffic and mulch decomposition provides nutrients to benefit the tree's roots and overall health.
A mulched area precludes the need for close maintenance. You don't need to walk under or into the branches to mow, rake or use weed cutting equipment to keep the garden looking its best. This prevents any collateral damage to the tree's trunk by cutting blades. Organic mulch diminishes need for intensive fertilization and irrigation.
Do not pile up mulch onto the cherry tree's trunk -- this leads to fungal problems on the trunk and root flare region. Tree roots extend well beyond the canopy. Extend and then maintain the mulch layer at least 2 feet beyond the branch tips where the smallest, newest roots reside as the tree matures.