Tree width (respectively, string path width) measures the number of partial (respectively, complete) computations of a nondeterministic finite automaton (NFA) on a given input. We characterize polynomial and exponential growth rates of tree width and string path width by structural properties of NFAs. Polynomial growth rates, roughly speaking, require that there exist computations going through a bounded number of cycles where the strings of characters labeling the cycles satisfy certain requirements. As the main result we show that the degrees of the polynomials bounding the tree width and string path width of an NFA differ from each other by at most one.

More generally, an NFA is said to have cycle height $\mathcal{K}$ if any computation can visit at most $\mathcal{K}$ distinct cycles. We give a polynomial time algorithm to decide whether an NFA has finite cycle height and, in the positive case, to compute its optimal cycle height.

树宽度（分别为字符串路径宽度）测量给定输入上非确定性有限自动机 (NFA) 的部分（分别为完整）计算的数量。我们通过 NFA 的结构特性来描述树宽度和字符串路径宽度的多项式和指数增长率。粗略地说，多项式增长率要求存在经过有限数量的循环的计算，其中标记循环的字符串满足某些要求。作为主要结果，我们表明，界定 NFA 的树宽度和字符串路径宽度的多项式的次数最多相差一个。

更一般地说，NFA 具有循环高度$\mathcal{ķ}$如果任何计算最多可以访问$\mathcal{ķ}$不同的周期。我们给出了一个多项式时间算法来确定 NFA 是否具有有限的循环高度，并在肯定的情况下计算其最佳循环高度。