Are there some types of dwarf crepe myrtles that bloom earlier than others? Do they bloom as early as the regular crepe myrtles? Will they bloom with only afternoon sun?

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There is a dwarf Crepe Myrtle called Early Bird that blooms about two months before the usual bloom start time. They will bloom with part shade but the flowers might be fewer - I had some that spent their lives under a giant California Pepper tree and they didn't seem to notice.

Daisy

There are three dwarf's that are said to bloom early:
Crepe Myrtle (Lagerstroemia Early Bird™ Purple)
Crepe Myrtle (Lagerstroemia Early Bird™ White)
Crepe Myrtle (Lagerstroemia Early Bird™ Lavender)