  In this paper, we study a new discrete tree and the resulting branching
process, which we call the \textbf{E}rlang \textbf{W}eighted
\textbf{T}ree(\textbf{EWT}). The EWT appears as the local weak limit of a
random graph model proposed in~\cite{La2015}. In contrast to the local weak
limit of well-known random graph models, the EWT has an interdependent
structure. In particular, its vertices encode a multi-type branching process
with uncountably many types.
  We derive the main properties of the EWT, such as the probability of
extinction, growth rate, etc. We show that the probability of extinction is the
smallest fixed point of an operator. We then take a point process perspective
and analyze the growth rate operator. We derive the Krein--Rutman eigenvalue
$\beta_0$ and the corresponding eigenfunctions of the growth operator, and show
that the probability of extinction equals one if and only if $\beta_0 \leq 1$.
