  We investigate the three-dimensional morphology of the dendrite tip using the
phase-field method. We find that, for low undercoolings, this morphology is
ostensibly independent of anisotropy strength except for a localized shape
distortion near the tip that only affects the value of the tip radius $\rho$
(which is crudely approximated by $\rho\approx (1-\alpha)\rho_{Iv}$ where
$\rho_{Iv}$ is the Ivantsov tip radius of an isothermal paraboloid with the
same tip velocity and $\alpha$ is the stiffness anisotropy). The universal tip
shape, which excludes this distortion, is well fitted by the form $z=-r^2/2+A_4
r^4\cos 4\phi$ where $|z|$ is the distance from the tip and all lengths are
scaled by $\rho_{Iv}$. This fit yields $A_4$ in the range $0.004-0.005$ in good
quantitative agreement with the existing tip morphology measurements in
succinonitrile [LaCombe et al., Phys. Rev. E {\bf 52}, 2778 (1995)], which are
reanalyzed here and found to be consistent with a single $\cos 4\phi$ mode
non-axisymmetric deviation from a paraboloid. Moreover, the fin shape away from
the tip is well fitted by the power law $z=-a |x|^{5/3}$ with $a\approx 0.68$.
Finally, the characterization of the operating state of the dendrite tip is
revisited in the light of these results.
