A syntax tree is a planar rooted tree where internal nodes are labeled on a graded set of generators. There is a natural notion of occurrence of contiguous pattern in such trees. We describe a way, given a set of generators $\mathfrak{G}$ and a set of patterns $\mathcal{P}$, to enumerate the trees constructed on $\mathfrak{G}$ and avoiding $\mathcal{P}$. The method is built around inclusion-exclusion formulas forming a system of equations on formal power series of trees, and composition operations of trees. This does not require particular conditions on the set of patterns to avoid. We connect this result to the theory of nonsymmetric operads. Syntax trees are the elements of such free structures, so that any operad can be seen as a quotient of a free operad. Moreover, in some cases, the elements of an operad can be seen as trees avoiding some patterns. Relying on this, we use operads as devices for enumeration: given a set of combinatorial objects we want enumerate, we endow it with the structure of an operad, understand it in term of trees and pattern avoidance, and use our method to count them. Several examples are provided.

语法树是一个平面根树，其中内部节点标记在一组分级的生成器上。在这样的树中自然会出现连续模式的概念。给定一组生成器，我们描述一种方法$\mathfrak{G}$ 和一组模式 $\mathcal{P}$，以枚举在 $\mathfrak{G}$ 并避免 $\mathcal{P}$。该方法建立在包含-排除公式的基础上，该公式形成了关于树的形式幂级数和树的合成运算的方程组。这不需要在模式集上避免特定的条件。我们将此结果与非对称操作符理论联系起来。语法树是这种自由结构的元素，因此任何操作数都可以看作是一个自由操作数的商。此外，在某些情况下，可以将操作员的元素视为避免某些模式的树。依靠此，我们将操作数用作枚举的设备：给定要枚举的一组组合对象，我们赋予它一个操作数的结构，从树和模式规避的角度理解它，并使用我们的方法对它们进行计数。提供了几个示例。