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Frangipanis fork naturally after they flower. Flower buds are produced at the branch tip and new tips are formed either side of the flower bud. Frangipani branches usually produce two new tips but small frangipani plants often just produce one tip beside a flower bud and some species can produce 3 or more new tips.
If you want your frangipani to produce more branches, the best thing to do is wait patiently for it to fork naturally. Natural forks have a nicer appearance and result in a better balanced frangipani tree.
When a frangipani branch is broken or cut off, the remaining branch stump will produce new shoots around the new branch end. Small and thin stumps often die but longer and thicker stumps will continue growing.
Forced forking often happens by accident or after pruning but it’s not a good way to get a frangipani to produce more branches.
New Branches From Trunk
Almost no frangipani species, pudica excepted, produce new branches from the trunk and along the branches. They only produce new branches after they flower and the branch tips.