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We introduce generalized dynamical pruning on rooted binary trees with edgelengths. The pruning removes parts of a tree $T$, starting from the leaves,according to a pruning function defined on subtrees within $T$. The generalizedpruning encompasses a number of discrete and continuous pruning operations,including the tree erasure and Horton pruning. The main result is invariance ofa finite critical binary Galton-Watson tree with exponential edge lengths withrespect to the generalized dynamical pruning for an arbitrary admissiblepruning function. The second part of the paper examines the continuum 1-Dballistic annihilation model $A+A \rightarrow \emptyset$ for a constantparticle density and initial velocity that alternates between the values of$\pm$1. The model evolution is equivalent to a generalized dynamical pruning ofthe shock tree that represents dynamics of sinks (points of particleannihilation), with the pruning function equal to the total tree length. Theshock tree is isometric to the level set tree of the model potential (integralof velocity). This equivalence allows us to construct a complete probabilisticdescription of the annihilation dynamics for the initial velocity thatalternates between the values of $\pm$1 at the epochs of a stationary Poissonprocess. Finally, we discuss several real tree representations of the ballisticannihilation model, closely connected to the shock wave tree.

Dynamical pruning of rooted trees with applications to 1D ballistic annihilation