Self-stabilizing and silent distributed algorithms for token distribution in rooted tree networks are given. Initially, each process of a graph holds at most $\ell$ tokens. Our goal is to distribute the tokens uniformly in the whole network so that every process holds exactly $k$ tokens. In the initial configuration, the total number of tokens in the network may not be $nk$ where $n$ is the number of processes in the network. The root process is given the ability to create a new token or remove a token from the network. We aim to minimize the convergence time, the number of token moves, and the space complexity. First, a self-stabilizing token distribution algorithm that converges within $O\left(n\ell \right)$ asynchronous rounds and needs $\Theta \left(nhϵ\right)$ redundant (or unnecessary) token moves is given, where $ϵ=min(k,\ell -k)$ and $h$ is the height of the tree network. Next, two novel mechanisms to reduce the number of redundant token moves are presented. One reduces the number of redundant token moves to $O\left(nh\right)$ without any additional costs while the other reduces the number of redundant token moves to $O\left(n\right)$, but increases the convergence time to $O\left(nh\ell \right)$. All given algorithms have constant memory at each process and each link register.

给出了在树状树网络中令牌分配的自稳定和静默分布式算法。最初，图的每个过程最多容纳$\ell$令牌。我们的目标是将令牌均匀分布在整个网络中，以便每个进程都能准确地容纳$ķ$令牌。在初始配置中，网络中的令牌总数可能不是$ñķ$ 哪里 $ñ$是网络中的进程数。根进程具有创建新令牌或从网络中删除令牌的能力。我们的目标是最小化收敛时间，令牌移动的数量和空间复杂度。首先，一种在内部收敛的自稳定令牌分配算法$Ø（ñ\ell ）$ 异步回合和需求 $\Theta （ñHϵ）$ 给出了多余的（或不必要的）令牌移动，其中 $ϵ=分（ķ，\ell -ķ）$ 和 $H$是树状网络的高度。接下来，提出了两种减少冗余令牌移动数量的新颖机制。一个减少了冗余令牌移动到的数量$Ø（ñH）$ 无需支付任何额外费用，而另一种则减少了向 $Ø（ñ）$，但将收敛时间增加到 $Ø（ñH\ell ）$。所有给定的算法在每个进程和每个链接寄存器中都有恒定的内存。