"Put out branches" = "Spread branches" = "Shoot out branches". Same meaning.
"Shoot out branches"과 "Spread branches"이 더 시적인 표현 같습니다.
Note: "spread branches" needs a determiner.
Wrong: "The tree spread branches." Right: "The tree spread its branches".
Branch out - 은유적인 표현입니다. Ex: "The company is doing well, but we need to branch out." "branch out" = "do something new"