bough, branch, twig, limb, counter for branches
枝 centers on the parts that grow out from a main trunk or stem: a branch, bough, twig, or limb. The counter sense is a natural extension for counting such offshoots.
枝 combines 木 (tree) with 支, which likely contributes both sound and the idea of branching or supporting. The character represents a tree's offshoots.
A tree 木 with a branching support 支: picture a tree trunk with limbs spreading out like supporting arms.
For シ, imagine a tree branch shaped like the letter 'C': C -> シ, and you count the C-shaped branches.
branch; bough; limb; twig; sprig; spray
edamame (green soybeans)
litchi (Nephelium litchi); lychee; lichee; litchi nut
toothpick
branches and leaves; foliage
ramification
twig; spray
weeping form (e.g. tree)
upper branches of a tree
lower branches of a tree
toothpick
transaction branch
bifurcation; branch; branching; branched chain; ramification
cuttings; lop; spray; set; slip
soursop (Annona muricata)
pruning (trees, shrubs, etc.)
large branch; bough
peppered moth (Biston betularia)
toothpick
pruning tree branches to encourage new growth