all $t\geq 0$, it holds that $$\begin{aligned}
        A(t)=\sum_{s\leq t}|\Delta_- \chi_{\emptyset}(s)|^{\omega_-}A_s\left((t-s)|\Delta_- \chi_\emptyset(s)|^{\alpha}\right),
    \end{aligned}$$ where $\Delta_-\chi_\emptyset(s):=\min\left\{0, \chi_\emptyset(s)-\chi_\emptyset(s-)\right\}$ (i.e. we only consider the negative jumps) and the $A_s$’s are i.i.d. copies of $A$ under $\mathcal{P}_1$, independent from $\chi_\emptyset$.

We first introduce a notation. If $\Delta_-\chi_\emptyset(s)\neq 0$ for some $s>0$, i.e. if the Eve cell gives birth at time $s$, we denote $\mathbb{U}_s$ the subtree generated by the newborn cell. We write $$\begin{aligned}
            A(t)
            &=\mathcal{A}\left(\left\{\ell\in\partial\mathbb{U}:\zeta_\ell\leq t\right\}\right)
            =\sum_{s\leq t}\mathds{1}_{\left\{\Delta_-\chi_{\emptyset}(s)\neq 0\right\}}\mathcal{A}\left(\left\{\ell\in\partial\mathbb{U}_s:\zeta_\ell\leq t\right\}\right).
        \end{aligned}$$ Thanks to , Lemma 3.2 of [@BBCK16] entails that the areas in the above sum are independent and that for all $s>0$ the conditional distribution of $\mathcal{A}\left(\left\{\ell\in\partial\mathbb{U}_s:\zeta_\ell\leq t\right\}\right)$ given $\chi_{\emptyset}$ is that of $A(t-s)$ under $\mathcal{P}_{|\Delta_-\chi_{\emptyset}(s)|}$, which by self-similarity is identical to that of $|\Delta_-\chi_{\emptyset}(s)|^{\omega_-}A((t-s)|\Delta_-\chi_{\emptyset}(s)|^{\alpha})$ under $\mathcal{P}_1$. (The time shift $-s$ comes from the fact that the root of $\mathbb{U}_s$ is at height $s$ in $\mathbb{U}$.)

We shall sometimes use Lemma \[Lemma Markov-branching property of A\] replacing $\chi_\emptyset(s)$ by $X(s)$, or equivalently $\exp(\xi_{\tau_s})$.

#### Choosing a spine according to the intrinsic area.

A classical tool in the study of branching processes is the so called *spinal decomposition* (see [@S15]). In [@BBCK16], the authors introduced one related to the intrinsic area (we refer to 