particles. If splitting took place at position $x$ then $$\tilde{P}(A(x) = n) = \left\{
    \begin{array}{rl}
    q_n& \text{if } x = 0, \\
    p_n& \text{if } x \neq 0. 
    \end{array} \right.$$

-   Uniformly at random one of the new particles is selected to continue the spine and thus to stochastically repeat the behaviour of the initial particle starting from $x$.

-   The remaining $A(x) - 1$ particles initiate independent copies of a branching process with branching rate $\beta_0 \delta_0(\cdot) + \beta$ and offsprng distribution $A(\cdot)$ as under $P^x$.

We shall now describe a family of martingale changes of measure that will put a certain bias on the motion of the spine particle as well as the birth rate and the offspring distribution along the path of the spine particle. Again, for a detailed discussion the reader is referred to [@HH09].

Let us consider a process of the form $$\begin{aligned}
\label{m_tilde}
\tilde{M}_t &= \Big[ \prod_{n=1}^{n_t} A_{n} \Big]  \mathrm{e}^{- \hat{\beta}t - \hat{\beta}_0 \tilde{L}_t} \tilde{M}^{(1)}_t \nonumber\\
&= \Big[ \prod_{n=1}^{n_t'} A_{n}' \times \prod_{n=1}^{n_t^0} A_{n}^0 \Big] \mathrm{e}^{- \hat{\beta}t - \hat{\beta}_0 \tilde{L}_t} \tilde{M}^{(1)}_t,\end{aligned}$$ where $(A_n)_{n \geq 1}$, $(A^0_n)_{n \geq 1}$, $(A_n')_{n \geq 1}$, $(n_t)_{t \geq 0}$, $(n^0_t)_{t \geq 0}$ and $(n_t')_{t \geq 0}$ were defined in Subsection 