fork of a tree, pitchfork
杈 centers on a forked shape, specifically a tree fork or a pitchfork-like tool. The two given meanings are direct applications of this branching form.
杈 combines 木 (tree) with 叉 (fork), suggesting a forked branch or a wooden pitchfork. The exact historical development is uncertain, but the component combination fits the meaning.
A tree 木 with a forked branch 叉: picture a wooden pitchfork made from a natural tree fork.
For サ, imagine a saw cutting a forked branch: saw -> サ, and the saw shapes the pitchfork.