Fan-in
Fan-in is the number of inputs a logic gate can handle. For instance the fan-in for the AND gate shown in the figure is 3. Physical logic gates with a large fan-in tend to be slower than those with a small fan-in. This is because the complexity of the input circuitry increases the input capacitance of the device. Using logic gates with higher fan-in will help reducing the depth of a logic circuit. Fan-in tree of a node refers to a collection of signals that contribute to the input signal of that  node. In quantum logic gates the fan-in always has to be equal to the number of outputs, the Fan-out. Gates for which the numbers of inputs and outputs differ would not be reversible (unitary) and are therefore not allowed.