branching process. Let $\mathbb{U}:=\bigcup_{n\geq 0}\mathbb{N}^n$ and let $\partial\mathbb{U}$ be the set of infinite sequences of natural integers. We shall also denote $\overline{\mathbb{U}}:=\mathbb{U}\cup\partial\mathbb{U}$. If $u\in\mathbb{N}^n$ for some $n\in\mathbb{N}$, we write $u(k)\in\mathbb{N}^k$ its ancestor at generation $k\leq n$ and $|u|=n$ its generation.

The process starts with a single cell that we call the Eve cell, indexed by $\emptyset$ and size at any time $t\geq 0$ denoted by $\chi_{\emptyset}(t)$. It evolves in time, starting from its birthtime $b_\emptyset:=0$ according to $\mathbb{P}_1$ until its absorption time $\zeta_\emptyset$ at 0. In this context, $\zeta_{\emptyset}$ is called the *lifetime* of $\chi_\emptyset$.

Since $\chi_\emptyset$ converges to 0 almost surely, it is possible to rank all its negative jumps in the decreasing order of the absolute values of their sizes. If $\left\{(b_i,\Delta_i);i\geq 1\right\}$ is the collection of times and sizes of these negative jumps ranked in this way, then for each $i\geq 1$, we start at time $b_i$ a new positive self-similar Markov process $\chi_i$ under the law $\mathbb{P}_{|\Delta_i|}$, independently of every other cell. We denote its lifetime $\zeta_i$, that is $\chi_i(t)$ is the size of the cell labelled $i$ if $b_i\leq t<b_i+\zeta_i$, and is sent to some cemetery state $\partial$ otherwise. In this manner, we construct recursively 