bud out
Of a plant, to produce buds (which blossom into leaves or flowers). I'm so relieved to see that flower budding out—I thought I had killed it!
a budding genius
A young person who exhibits signs of great intelligence. This kid's a budding genius—have you seen the complex equations he can solve?
bud out
[for a flowering plant or tree] to develop buds. How early in the spring do the trees bud out around here? The trees bud out in early spring.
budding genius
a very bright and promising young person. Harry is a budding genius, but he seems like a fairly normal teenager.