The Wiener polarity index $W_p$ is a topological index that was devised by the chemist Harold Wiener for predicting the boiling points of alkanes. The index $W_p$ for chemical trees (chemical graphs representing alkanes) is defined as the number of unordered pairs of vertices at distance 3. A vertex of a chemical tree with degree at least 3 is called a branching vertex. A segment of a chemical tree $T$ is a nontrivial path $S$ whose end-vertices have degrees different from 2 in $T$ and every other vertex (if exists) of $S$ has degree 2 in $T$. In this paper, sharp upper and lower bounds on the Wiener polarity index $W_p$ are derived for the chemical trees of a fixed order and with a given number of branching vertices or segments, and for every such bound, a class of trees attaining that bound is obtained. As a consequence of the derived results, a vital step towards the complete solution of an existing open problem concerning the maximum $W_p$ value of chemical trees is provided.

维纳极性指数 ${\mathrm{w\; ^}}_p$是由化学家Harold Wiener设计的拓扑指数，用于预测烷烃的沸点。指标${\mathrm{w\; ^}}_p$用于化学树的化学词（代表烷烃的化学图）定义为距离3处无序顶点对的数量。化学树的顶点度至少为3的顶点称为分支顶点。化学树的一部分$Ť$ 是一条不平凡的道路 $\mathrm{小号}$ 其最终顶点的度数不同于2 in $Ť$ 和每个其他顶点（如果存在） $\mathrm{小号}$ 拥有2级学位 $Ť$。在本文中，维纳极性指数的上下界清晰${\mathrm{w\; ^}}_p$对于固定顺序的化学树，使用给定数量的分支顶点或分段，可以从生成树。并且对于每个这样的界线，都会获得一类达到该界线的树。作为得出结果的结果，朝着彻底解决现有最大问题的关键一步迈出了重要一步。${\mathrm{w\; ^}}_p$ 提供了化学树的价值。