As a classifier, 枝 is used for flowers (一枝花), pens (一枝笔), and similar long, thin objects. It's interchangeable with 支 (zhī) as a classifier in many cases.
memory
木 (wood) + 支 (branch off) - a wooden thing that branches off from the trunk.
木 is the tree radical and locks 枝 into the wood family - a branch is literally a part of a tree. Same indexing in 树 tree, 林 forest, 根 root. The 木 here keeps its full four-stroke form on the left.
支 supplies the sound zhī and reinforces meaning - 支 itself depicts a hand holding a branch and means "branch off, prop up." So 枝 is one of the rare phono-semantic compounds where the phonetic doubles as a semantic clue.
