We are trying to find the average sum of branches on a tree representing a birth-death process, with fixed birth and death parameters.
Let $X_t$ be the number of individuals at time $t\geq 0$, with $X_0=0$. Suppose we stop the process when $X_t=n$. This defines a stopping time $T$, and hence, if $T$ is finite, we have to compute the integral
$$E[\int_{0}^{T}X_t dt]$$
Since the integral goes up to a stopping time, I don't think I can exchange the expectation and the integral. Do you know how to solve this problem?