Simply Because You Care
The Hebrew word for olive tree is "es shemen," which literally means "tree of oil." It is from the primitive root meaning "to shine." It means richness, annointing, fruitful, ointment, olive. It is related to the word "shemesh," to be brilliant, and which also is the Hebrew word for the sun, the giver of life, and warmth, brightly shining in the sky. Giving an olive branch says "I think you shine, you radiate, you are brilliant."