I am working on a paper of Elie Aidekon : *‘Speed of the biased random walk on a Galton–Watson tree’* and have a question about one transformation in a proof:
\begin{align}
& 1+\frac{1}{1-\lambda}+\mathbb{E} \biggl[ \frac{\beta_n(I) \mathbf{1}_{\{ \beta(j)=0 \ \forall j\neq I\}}}{\lambda-1+\sum_{i=1}^{\nu(e)}\beta_n(i)}\biggl] \\
\tag{1}
\label{1}
={} & \frac{\lambda}{1-\lambda}+\mathbb{E} \biggl[ \frac{\beta_n(I) \mathbf{1}_{\{ \beta(j)=0 \ \forall j\neq I\}}}{\lambda-1+\beta_n(I)}\biggl] \\
\tag{2}
\label{2}
={} &\frac{\lambda}{1-\lambda}+\mathbb{E} [\nu q^{\nu-1}] \mathbb{E} \biggl[ \frac{\beta_{n-1}(e)}{\lambda-1+\beta_{n-1}(e)}\biggl].
\end{align}

Some additional informations about it:

It holds that $\lambda<1$,

$\beta(e)=\mathcal{P}^{e}(\tau_{e_{*}}=\infty)$ is the probability that the random walk start in the root $e$ of a tree and never hits the parent $e_*$,

$\beta_n(e)=\mathcal{P}^{e}\bigl(\tau^{(n)}<\tau_{e_*}\bigl)$ is the probability that the random walk start in the root $e$ of a tree and hit level $n$ before it hit the parent $e_*$.

I absolutely don’t know where the $\frac{\lambda}{1-\lambda}$ in the \eqref{1} equation comes from and how the denominator short up to $\lambda-1+\beta_n(I)$. Have someone a tip for me?

In the \eqref{2} equation I know that the *branching property* is used, but I don’t see this. I just know the branching property defined by ‘lines’ and can’t see how to take this definition for markov chains (without lines).