The spectrum of BPS states in type IIA string theory compactified on a Calabi-Yau threefold famously jumps across codimension-one walls in complexified Kähler moduli space, leading to an intricate chamber structure. The Split Attractor Flow Conjecture posits that the BPS index \Omega_z(\gamma) for given charge \gamma and moduli z can be reconstructed from the attractor indices \Omega_*(\gamma_i) counting BPS states of charge \gamma_i in their respective attractor chamber, by summing over a finite set of decorated rooted flow trees known as attractor flow trees. If correct, this provides a classification (or dendroscopy) of the BPS spectrum into different topologies of nested BPS bound states, each having a simple chamber structure. Here we investigate this conjecture for the simplest, albeit non-compact, Calabi-Yau threefold, namely the canonical bundle over P^2. Since the Kähler moduli space has complex dimension one and the attractor flow preserves the argument of the central charge, attractor flow trees coincide with scattering sequences of rays in a two-dimensional slice of the scattering diagram in the space of stability conditions on the derived category of compactly supported coherent sheaves on K_{P^2}. We combine previous results on the scattering diagram of K_{P^2} in the large volume slice with new results near the orbifold point \mathbb{C}^3/\mathbb{Z}_3, and argue that the Split Attractor Flow Conjecture holds true on the physical slice of \Pi-stability conditions. In particular, while there is an infinite set of initial rays related by the group \Gamma_1(3) of auto-equivalences, only a finite number of possible decompositions \gamma=\sum_i\gamma_i contribute to the index \Omega_z(\gamma) for any \gamma and z, with constituents \gamma_i related by spectral flow to the fractional branes at the orbifold point.
