Spine results and applications
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Spine construction
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In this section we extend our probability space by introducing the spine process. A more detailed description of this procedure may be found for example in [@HH09].

The spine of the branching process, which we shall denote by $\xi$, is the infinite line of descent of particles chosen uniformly at random from all possible lines of descent. It is constructed in the following way. The initial particle $\varnothing$ of the branching process begins the spine. When the initial particle dies and is replaced with a random number of new particles, one of them is chosen uniformly at random to continue the spine. This procedure is then repeated recursively: whenever the particle in the spine dies, one of its children is chosen uniformly at random to continue the spine. We may then write the spine as $\xi = \{ \xi_0, \xi_1, \xi_2, \cdots\}$, where $\xi_n$ is the lable of the spine particle in the $n$th generation and $\xi_0 = \varnothing$.

We let $\tilde{P}$ denote the probabiity measure under which the branching process is defined together with the spine. Hence $P = \tilde{P}|_{\mathcal{F}_\infty}$. We let $\tilde{E}$ be the expectation corresponding to $\tilde{P}$.

Below introduce 