We have systematically characterized the degradation of imaging quality with depth in deep brain multi-photon microscopy, utilizing our recently developed numerical model that computes wave propagation in scattering media. The signal-to-background ratio (SBR) and the resolution determined by the width of the point spread function are obtained as functions of depth. We compare the imaging quality of two-photon (2PM), three-photon (3PM), and non-degenerate two-photon microscopy (ND-2PM) for mouse brain imaging. We show that the imaging depth of 2PM and ND-2PM are fundamentally limited by the SBR, while the SBR remains approximately invariant with imaging depth for 3PM. Instead, the imaging depth of 3PM is limited by the degradation of the resolution, if there is sufficient laser power to maintain the signal level at large depth. The roles of the concentration of dye molecules, the numerical aperture of the input light, the anisotropy factor g, noise level, input laser power, and the effect of temporal broadening are also discussed.