Representations of regular trees and invariants of AR-components for generalized Kronecker quivers

We investigate the generalized Kronecker algebra $\mathcal{K}_r = k\Gamma_r$ with $r \geq 3$ arrows. Given a regular component $\mathcal{C}$ of the Auslander-Reiten quiver of $\mathcal{K}_r$, we show that the quasi-rank $rk(\mathcal{C}) \in \mathbb{Z}_{\leq 1}$ can be described almost exactly as the distance $\mathcal{W}(\mathcal{C}) \in \mathbb{N}_0$ between two non-intersecting cones in $\mathcal{C}$, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality $$-\mathcal{W}(\mathcal{C}) \leq rk(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.$$ Utilizing covering theory, we construct for each $n \in \mathbb{N}_0$ a bijection $\varphi_n$ between the field $k$ and $\{ \mathcal{C} \mid \mathcal{C} \ \text{regular component}, \ \mathcal{W}(\mathcal{C}) = n \}$. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.