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Monadic Second-Order Classes of Forests with a Monadic Second-Order 0-1 Law

Let $\cT$ be a monadic-second order class of finite trees, and let $\bT(x)$ be its (ordinary) generating function, with radius of convergence $ρ$. If $ρ\ge 1$ then $\cT$ has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators $(n)$ and $(\ge n)$. Using this, one has an explicit expression for $\bT(x)$ in terms of the initial functions $x$ and $x\cdot \big(1-x^n\big)^{-1}$, the operations of addition and multiplication, and the Pólya exponentiation operators $\sE_n, \sE_{\ge n}$. Let $\cF$ be a monadic-second order class of finite forests, and let $\bF(x)=\sum_n f(n) x^n$ be its (ordinary) generating function. Suppose $\cF$ is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class $\cT$ of trees in $\cF$, Compton's theory of 0--1 laws, and a significantly strengthened version of 2003 results of Bell and Burris on generating functions, we show that $\cF$ has a monadic second-order 0--1 law iff the radius of convergence of $\bF(x)$ is 1 iff the radius of convergence of $\bT(x)$ is $\ge 1$.