Given an $\mathbb{N}$-weighted tree automaton, we give a decision procedure for exponential vs polynomial growth (with respect to the input size) in quadratic time, and an algorithm that computes the exact polynomial degree of growth in cubic time. As a special case, they apply to the growth of the ambiguity of a nondeterministic tree automaton, i.e. the number of distinct accepting runs over a given input. We deduce analogous decidability results (ignoring complexity) for the growth of the number of results of set queries in Monadic Second-Order logic (MSO) over ranked trees. In the case of polynomial growth of degree $k$, we also prove a reparameterization theorem for such queries: their results can be mapped to $k$-tuples of input nodes in a finite-to-one and MSO-definable fashion. We then apply these tools to study growth rates and subclass membership problems for tree-to-tree functions. Using new proof strategies, we recover and generalize known results concerning polyregular functions, total deterministic macro tree transducers, and partial nondeterministic top-down tree transducers. In particular, we give a procedure to decide polynomial size-to-height increase for both macro tree transducers and MSO set interpretations, and compute the degree. The paper concludes with a survey of a wide range of related work.

翻译：给定一个$\mathbb{N}$带权树自动机，我们提出一个在二次时间内判定其增长为指数型或多项式型（相对于输入规模）的决策过程，以及一个在三次时间内精确计算多项式增长次数的算法。作为特例，这些方法适用于非确定性树自动机的歧义性增长问题，即对给定输入的相异接受运行路径数量。我们推导出在秩化树上关于Monadic Second-Order逻辑（MSO）集合查询结果数量增长的类似可判定性结果（忽略复杂度）。对于次数为$k$的多项式增长情形，我们还证明了此类查询的重参数化定理：其查询结果可通过有限对一且MSO可定义的方式映射至输入节点的$k$元组。随后，我们运用这些工具研究树到树函数的增长率及子类归属问题。通过采用新的证明策略，我们恢复并推广了关于多正则函数、完全确定性宏树转换器以及部分非确定性自顶向下树转换器的已知结果。特别地，我们针对宏树转换器与MSO集合解释两种情况，给出了判定多项式规模到高度增长的决策过程并计算其次数。本文最后对大量相关研究工作进行了综述。