The growing dendrites during the charging of the battery can pierce into the polymer electrolyte and short-circuit the cell. A dendritic microstructure is a branched medium, consisting of numerous three-way intersections and when subjected to the piercing pressure the corresponding forces propagate throughout the dendrite body. In this paper, we explore the effect of such force propagation versus the geometry of a three-way intersection leading to the failure of a dendritic branch and we have analyzed the dominant failure mechanism in the branch. Particularly we address the mechanics of the intersection versus the respective inclinations of the branches and their normalized length. Generalizing this method into larger clusters of dendritic trees helps to identify the dominant failure mechanism and devise methods to prevent it.