of our branching process in this respect is the following Doney condition:

**Assumption $A1$**. There exists a number $0<\rho <1$ such that $$\mathbf{P}(S_{n}>0)\rightarrow \rho \text{ as }n\rightarrow \infty .$$

As it was shown in [@Do], Condition **$A1$** is equivalent to the classical Spitzer condition $$\frac{1}{n}\sum_{k=1}^{n}\mathbf{P}(S_{k}>0)\rightarrow \rho \text{ as }n\rightarrow \infty .$$Recall that Assumption **$A1$** implies $$n^{-1}\tau \left( n\right) \overset{d}{\rightarrow }\tau ,\quad n\rightarrow
\infty ,  \label{arcsin}$$where $\tau $ is a random variable distributed according to the generalized arcsine law with parameter $\rho $ ([@S], Ch. IV, § 20) and the symbol $\overset{d}{\rightarrow }$ means convergence in distribution.

 Vatutin and Dyakonova, using the quenched approach, have proved in [VD3]{} and [@VD05] conditional limit theorems given $\left\{ T>n\right\} $ describing the asymptotic behavior, as $n\rightarrow \infty ,$ of the distribution of the number of particles at moments $Z_{nt}$ and $Z_{\tau
(nt)},t\in (0,1),~$ in the branching processses in random environment respecting Assumption $A1$.

In the present paper we consider the conditioning $\left\{ T=n\right\} $ and, under Assumption A1 and the quenched approach, study the distribution of the number of particles in our branching process either at moments $nt,$ $t\in (0,1),$ or at moments located in a vicinity of points $\tau (nt)$ or $\tau (nt,n)$. 