Branching processes, random trees, and a generalized scheme of arrangements of particles

It is shown that the conditional distributions of a number of characteristics of a branching process $\mu(t)$, $\mu(0)=m$, under the condition that the number of total progeny $\nu_m$ in this process is equal to $n$, coincide with the distributions of the corresponding characteristics of a generalized scheme of arrangement of particles in cells. In the case where the number of offsprings of a particle has the Poisson distribution, the characteristics of the branching process $\mu(t)$, $\mu(0)=1$, under the condition that $\nu_1=n+1$, coincide with the characteristics of a random tree. By using these connections we obtain in this article a series of limit theorems as $n\to\infty$ for characteristics of random trees and branching processes under the conditions that $\nu_m=n$.