Eureka lemons are one of the two "true" types of lemons (the other being Lisbon lemons), and the plants originated in California. Eureka plants are thin and wide-spreading, with sparse branch growth. Most citrus trees bear fruit during the fall and winter, but Eureka plants bear fruit during the spring and summer. Proper pruning encourages new growth.
Clip or pinch off any dead or yellow leaves. These do not affect the health of the plant, but pinching them off makes the plant more visually appealing.
Cut off any dead branches. Use sharp gardening shears to make a clean, smooth cut. Remove thin and sickly branches. These will never produce fruit, but will continue to sap energy away from the plant as long as they are intact.
Trim some of the height to keep it more manageable if the plant is taller than you'd like. Cut off long branches that extend beyond the rest of the plant. Give the plant a full, rounded shape.
Check fruit clusters, and remove any runt fruits that are among the clusters. Improperly developing fruits affect the development and flavor of other fruits in their clusters.