peduncle, stem of plants
蒂 centers on the connecting stalk of a plant: the peduncle or stem that attaches a flower, fruit, or leaf to the main body.
蒂 combines 艹 (grass/plant) with 帝, which likely contributes the sound and may relate to a base or foundation, suggesting the part of a plant that connects to the main stem.
The grass top 艹 sits above 帝, like a plant's stem holding up a flower or fruit. Picture the 帝 part as a sturdy base from which the plant part hangs.
For テイ, imagine a tea stem: the word 'tea' sounds like テイ, and you picture the stem of a tea plant holding a leaf.
stem and calyx of muskmelon (used in traditional Chinese medicine)