Embedded trees are labelled rooted trees, where the root has zero label and where the labels of adjacent vertices differ (at most) by $\pm 1$. Recently it has been proved (see Chassaing and Schaeffer (2004) [8] and Janson and Marckert (2005) [11]) that the distribution of the maximum and minimum labels are closely related to the support of the density of the integrated superbrownian excursion (ISE). The purpose of this paper is to make this probabilistic limiting relation more explicit by using a generating function approach due to Bouttier et al. (2003) [6] that is based on properties of Jacobi’s $\theta$-functions. In particular, we derive an integral representation of the joint distribution function of the supremum and infimum of the support of the ISE in terms of the Weierstrass $\wp$-function. Furthermore we re-derive the limiting radius distribution in random quadrangulations (by Chassaing and Schaeffer (2004) [8]) with the help of exact counting generating functions.

嵌入式树被标记为根树，其中根的标签为零，相邻顶点的标签相差（最多）为 $\pm \mathrm{1个}$。最近，已经证明（参见Chassaing和Schaeffer（2004）[8]以及Janson和Marckert（2005）[11]），最大和最小标签的分布与集成超布朗偏移密度的支持密切相关。 （ISE）。本文的目的是通过使用Bouttier等人的生成函数方法使这种概率限制关系更加明确。（2003）[6]是基于Jacobi的性质$\theta$-功能。特别是，我们根据Weierstrass推导了ISE的最高和最低支持的联合分布函数的积分表示。$\wp$-功能。此外，借助精确计数生成函数，我们重新推导了随机四边形中的极限半径分布（由Chassaing和Schaeffer（2004）[8]）。