In this paper we study a special subclass of monotonically labeled increasing trees: each sequence of labels from the root to any leaf is strictly increasing and each integer between $1$ and $k$ must appear in the tree, where $k$ is the largest label. The main difference with the classical model of binary increasing tree is that the same label can appear in distinct branches of the tree. Such a class of trees can be used in order to model population evolution processes or concurrent processes. A specificity of such trees is that they are built through an evolution process that induces ordinary generating functions. Finally, we solve the nice counting problem for these trees of size $n$ and observe interesting asymptotics involving powers of $n$ with irrational exponents.

中文翻译：

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关于标签重复增加的树的数量

在本文中，我们研究了单调标记递增树的一个特殊子类：从根到任何叶子的每个标签序列都是严格递增的，并且 $1$ 和 $k$ 之间的每个整数都必须出现在树中，其中 $k$ 是最大的标签。与二叉递增树的经典模型的主要区别在于，相同的标签可以出现在树的不同分支中。此类树可用于对种群进化过程或并发过程进行建模。这种树的一个特殊性是它们是通过一个进化过程构建的，该过程引发了普通的生成函数。最后，我们为这些大小为 $n$ 的树解决了很好的计数问题，并观察了有趣的渐近性，涉及 $n$ 的幂和无理指数。