From the work of Erd\H{o}s and R\'{e}nyi from 1963 it is known that almost all graphs have no symmetry. In 2017, Lupini, Man\v{c}inska and Roberson proved a quantum counterpart: Almost all graphs have no quantum symmetry. Here, the notion of quantum symmetry is phrased in terms of Banica's definition of quantum automorphism groups of finite graphs from 2005, in the framework of Woronowicz's compact quantum groups. Now, Erd\H{o}s and R\'{e}nyi also proved a complementary result in 1963: Almost all trees do have symmetry. The crucial point is the almost sure existence of a cherry in a tree. But even more is true: We almost surely have two cherries in a tree - and we derive that almost all trees have quantum symmetry. We give an explicit proof of this quantum counterpart of Erd\H{o}s and R\'{e}nyi's result on trees.

中文翻译：

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几乎所有的树都具有量子对称性

从 1963 年 Erd\H{o}s 和 R\'{e}nyi 的工作可知，几乎所有的图都没有对称性。2017 年，Lupini、Man\v{c}inska 和 Roberson 证明了量子对应：几乎所有图都没有量子对称性。在这里，在 Woronowicz 紧致量子群的框架内，量子对称性的概念是根据 Banica 2005 年对有限图的量子自同构群的定义来表述的。现在，Erd\H{o}s 和 R\'{e}nyi 在 1963 年也证明了互补的结果：几乎所有的树都具有对称性。关键的一点是几乎可以肯定地存在一棵树上的樱桃。但更真实的是：我们几乎可以肯定，一棵树上有两个樱桃——我们推导出几乎所有的树都具有量子对称性。我们给出了 Erd\H{o}s 和 R\'{e}nyi 在树上的这个量子对应物的明确证明。