  In this paper, a general tree algorithm processing a random flow of arrivals
is analyzed. Capetanakis--Tsybakov--Mikhailov's protocol in the context of
communication networks with random access is an example of such an algorithm.
In computer science, this corresponds to a trie structure with a dynamic input.
Mathematically, it is related to a stopped branching process with exogeneous
arrivals (immigration). Under quite general assumptions on the distribution of
the number of arrivals and on the branching procedure, it is shown that there
exists a positive constant $\lambda_c$ so that if the arrival rate is smaller
than $\lambda_c$, then the algorithm is stable under the flow of requests, that
is, that the total size of an associated tree is integrable. At the same time,
a gap in the earlier proofs of stability in the literature is fixed. When the
arrivals are Poisson, an explicit characterization of $\lambda_c$ is given.
Under the stability condition, the asymptotic behavior of the average size of a
tree starting with a large number of individuals is analyzed. The results are
obtained with the help of a probabilistic rewriting of the functional equations
describing the dynamics of the system. The proofs use extensively this
stochastic background throughout the paper. In this analysis, two basic limit
theorems play a key role: the renewal theorem and the convergence to
equilibrium of an auto-regressive process with a moving average.
