Dynamical Pruning of Rooted Trees with Applications to 1-D Ballistic Annihilation

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 $A+A\to \varnothing$ for a constant particle density and initial velocity that alternates between the values of $\pm 1$. 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.