A compacted binary tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special class of directed acyclic graphs. We are interested in the asymptotic number of compacted trees of given size, where the size of a compacted tree is given by the number of its internal nodes. Due to its superexponential growth this problem poses many difficulties. Therefore we restrict our investigations to compacted trees of bounded right height, which is the maximal number of edges going to the right on any path from the root to a leaf.

We solve the asymptotic counting problem for this class as well as a closely related, further simplified class.

For this purpose, we develop a calculus on exponential generating functions for compacted trees of bounded right height and for relaxed trees of bounded right height, which differ from compacted trees by dropping the above described uniqueness condition. This enables us to derive a recursively defined sequence of differential equations for the exponential generating functions. The coefficients can then be determined by performing a singularity analysis of the solutions of these differential equations.

Our main results are the computation of the asymptotic numbers of relaxed as well as compacted trees of bounded right height and given size, when the size tends to infinity.

压缩的二叉树是从二叉树创建的图，这样原始树中重复出现的子树由指向现有树的指针表示，因此每个子树都是唯一的。这样的表示形成一类特殊的有向无环图。我们对给定大小的压缩树的渐近数量感兴趣，其中，压缩树的大小由其内部节点的数量给出。由于其超指数增长，这个问题带来了许多困难。因此，我们将研究限制在高度合适的紧实树上，该树是从根到叶的任何路径上向右的最大边数。

我们为该类以及一个紧密相关的，进一步简化的类解决渐近计数问题。

为此，我们针对有界右高的压缩树和有界右高的松弛树的指数生成函数开发了一种演算，它们通过删除上述唯一性条件而与压缩树不同。这使我们能够为指数生成函数导出递归定义的微分方程序列。然后可以通过对这些微分方程解的奇异性分析来确定系数。

我们的主要结果是，当大小趋于无穷大时，计算有限高度和给定大小的松弛树和压实树的渐近数。