We examine a discrete random recursive tree growth process that, at each time step, either adds or deletes a node from the tree with fixed, complementary probabilities. Node addition follows the usual uniform attachment model. For node removal, we identify a class of deletion rules guaranteeing the current tree conditioned on its size is uniformly distributed over its range. By using generating function theory and singularity analysis, we obtain asymptotic estimates for the expectation and variance of a tree’s size, as well as its expected leaf count and root degree. In all cases, the behavior of such trees falls into three regimes determined by the insertion probability. Interestingly, the results are independent of the specific class member deletion rule used.

我们研究了一个离散随机递归树生长过程，该过程在每个时间步中都以固定的互补概率从树中添加或删除一个节点。节点添加遵循通常的统一附件模型。对于节点删除，我们确定了一类删除规则，以保证以其大小为条件的当前树在其范围内均匀分布。通过使用生成函数理论和奇异性分析，我们获得了树大小的期望和方差，以及其期望叶数和根度的渐近估计。在所有情况下，这些树的行为分为由插入概率决定的三个机制。有趣的是，结果与使用的特定类成员删除规则无关。