In early spring in the woods around my house,the ground is covered solid with thousands of trout-lilly plants,but they only bloom heavily along wet spots in the deep shade.I guess they like moist, rich soil,where they can take advantage of the early spring light before the ground becomes shaded by the developing leaves of the trees above them.They transplant easly.

My understanding with trout lilies is that it takes them many years to mature, and when they do you will have two leaves from the base instead of one. At this point they should start to flower.

Likely more sun. They are bulbs and only get sunlight for about 4-6 weeks in early spring, in that time they have to build up sufficient food in the bulb to support two leaves and a flower. It can take several springs to build up enough reserve in the bulb, until then they only put out one leaf and don't bloom.

The mottled leaves are attractive, even without flowers.

I assume you have the yellow-flowering Erythronium americanum, which is notoriously shy flowering. Location is the key to more flower production. Flood plains under deciduous trees is the most productive. Moist, upland deciduous forests that are gently rolling or flat will only yield 5-10% flowering among the mature plants. Those on steep slopes will rarely, if ever, bloom.
E. americanum requires 8-9 years to reach maturity from seed or daughter corms that are formed at the end of underground stolons, which is one of the primary methods of forming colonies. Substantial corm growth occurs during the 4th and 5th year of development. Underground growth of plants begin in the fall and continues during winter months and uptake of nutrients, principally Nitrogen and Potassium, peak just before the foliage and flower stem emerge in the spring. They require months of cold temperatures to thrive. I manage to keep a few alive here, just to see if it's possible. Our local yellow-flowering natives are E. umbilicatum, ssp. umbilicatum and there's nothing shy about them! I have found them growing and profusely flowering on very steep, rocky slopes, seemingly, a preferred habitat.
Rb

Thanks RP.

The plants (americanum) have been undisturbed for many years, so many must be mature. If they like more water, then this spring should be a very good year for them since this fall has been so wet.

Based on the above info, water and fertilizer in the fall might improve flowering.