operator on trees and quantum graphs, see [@km]. However, it seems that resonances have not been systematically studied in the context of (regular) trees.

In this paper, we use resonance methods to obtain better understanding of local spectral properties for perturbed Schrödinger operators on regular tree graphs with $k$-fold branching, $k \geq 1$, as we describe below (cf. Section \[secd\]). Our techniques are similar to those used in [@bbr1; @bbr2] (and references therein), where self-adjoint perturbations are considered. Actually, these methods can be extended to nonself-adjoint models, see for instance [@sa]. Here, we are focused on some nonself-adjoint perturbations of the Laplace operator on regular tree graphs. In particular, we shall derive as a by-product, a description of the eigenvalues distribution near the spectral thresholds of the operator.

Since a nonself-adjoint framework is involved in this article, it is convenient to clarify the different notions of spectra we use. Let $T$ be a closed linear operator acting on a separable Hilbert space $\mathscr{H}$, and $z$ be an isolated point of $\sigma(T)$ the spectrum of $T$. If $\gamma$ is a small contour positively oriented containing $z$ as the only point of $\sigma(T)$, we recall that the Riesz projection $P_z$ associated to 