Formulas vs Circuits
No. Take for example $\mathsf{AC^0}$. It is polynomial-size bounded-depth circuits. Being DAG or tree doesn't change the class. The fan-in can be arbitrarily large. 

If the fan-in is bounded we end-up with $\mathsf{NC^0}$ which is much weaker. It cannot compute any function where the output depends on more than a bounded number of inputs as each gate with at least two inputs will decrease the number of usable wires by one and there is no way of increasing them.

Note also that the number of gates (ignoring single input gates like negation) in any circuit with $n$ inputs and with fan-out $1$ is going to be at most $n-1$.