We introduce generalized dynamical pruning on rooted binary trees with edge lengths that encompasses a number of discrete and continuous pruning operations, including the tree erasure and Horton pruning. The pruning removes parts of a tree T , starting from the leaves, according to a pruning function defined on descendant subtrees within T . We prove the invariance of critical binary Galton–Watson tree with exponential edge lengths with respect to the generalized dynamical pruning for an arbitrary admissible pruning function. These results facilitate analysis of the continuum 1-D ballistic annihilation model $$A+A \rightarrow \varnothing $$ A + A → ∅ for a constant particle density and initial velocity that alternates between the values of $$\pm 1$$ ± 1 . We show that the model’s shock wave is isometric to the level set tree of the potential function, and the model evolution is equivalent to the generalized dynamical pruning of the shock wave tree.

中文翻译：

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有根树的动态修剪与一维弹道歼灭的应用

我们在边缘长度包含许多离散和连续修剪操作的有根二叉树上引入广义动态修剪，包括树擦除和霍顿修剪。根据在 T 内的后代子树上定义的修剪函数，修剪从叶子开始移除树 T 的部分。我们证明了具有指数边长的临界二元 Galton-Watson 树相对于任意可容许剪枝函数的广义动态剪枝的不变性。这些结果有助于分析连续一维弹道湮灭模型 $$A+A \rightarrow \varnothing $$ A + A → ∅ 对于恒定的粒子密度和初始速度在 $$\pm 1$$ 值之间交替± 1 .