A patch beneath a tree is a different growing space from the sunny middle of your yard. Start by deciding whether you need walkable turf there, or simply an attractive surface. That choice can save a cycle of reseeding and disappointment.
Map the light, then the traffic
Photograph the patch in the morning, at midday and in the afternoon while the tree is in leaf. Mark the route people and pets actually take. Penn State describes both limited light and competition from trees as obstacles to turf. Seed cannot remove either constraint.
Give suitable grass a fair trial
Identify your existing grass, then ask your local extension service which shade-tolerant options fit your climate. Reduce unnecessary traffic and avoid automatically giving the shaded patch the same irrigation as full sun. Keep a small trial area so you can compare its density through the growing season.
Know when to change the surface
Where light stays inadequate, sketch a bed or a suitable shade groundcover instead. Keep soil and mulch away from the trunk flare, and avoid cutting large roots or burying them to create a smooth lawn. A tree-care professional can assess root or canopy work. The goal is a deliberate edge between lawn and tree area, not turf at any cost.
Do not assume a bag labeled “shade mix” suits every state or every level of shade.