Trees can be stressed by flowers that need lots of water or have large root systems. Drought-resistant plants that have shallow roots and require small planting holes are the best choices. To avoid damaging roots, the Million Trees NYC website says not to plant within a foot of the trunk.
Annuals that can tolerate shade and drier conditions include sweet alyssum (Lobularia maritima), lobelia (Lobelia inflata), nasturtium (Tropaeolum majus) and salvia (Salvia officinalis). However, the John Deere website notes that planting annuals yearly under trees increases the risk of root damage, because many trees have roots that spread laterally. It says to try perennials and ground covers instead.
Plantings under certain trees, such as evergreens and some oaks, need to tolerate acid soil. Hostas are acid-loving perennials with small root masses that tolerate limited moisture and shade. The Garden Lady website also recommends woodland flowers such as celandine poppy (Stylophorum diphyllum).