Given a connected graph $G=(V_G,E_G)$ with $x,y\in V_G,$ the hitting time $H_G(x,y)$ is defined as the expected number of steps that a simple random walk takes to go from x to y. A hitting-time-based invariant, called the ZZ index, was first proposed by Zhu and Zhang [The hitting time of random walk on unicyclic graphs. Linear Multilinear Algebra. doi:10.1080/03081087.2019.1611732] recently. It was defined to be $\psi \left(G\right)=\underset{x,y\in V_G}{max}{H}_G(x,y).$ In this paper, some extremal problems on the ZZ index of trees with some given parameters are considered. Firstly, sharp upper and lower bounds on $\psi \left(G\right)$ are determined, respectively, for trees with given diameter, matching number, given vertex bipartition and given pendant numbers. Secondly, ordering the n-vertex caterpillar trees with respect to their ZZ indices are established.

给定一个连通图$G=({五}_G,{乙}_G)$和$X,\mathrm{是的}\in {五}_G,$击球时间$H_G(X,\mathrm{是的})$定义为简单随机游走从x到y的预期步数。一个基于命中时间的不变量，称为ZZ指数，由朱和张首先提出 [The hit time of random walk on unicyclic graphs。线性多线性代数。doi:10.1080/03081087.2019.1611732] 最近。它被定义为$\psi \left(G\right)=\underset{X,\mathrm{是的}\in {五}_G}{最大限度}{H}_G(X,\mathrm{是的}).$在本文中，考虑了一些给定参数的树的ZZ索引的极值问题。首先，明确的上限和下限$\psi \left(G\right)$分别针对具有给定直径、匹配数、给定顶点二分和给定垂饰数的树确定。其次，建立了关于它们的ZZ索引的n顶点毛虫树的排序。