  We use a phase-field model to study the effect of confinement on dendritic
growth, in a pure material solidifying in an undercooled melt, and in the
directional solidification of a dilute binary alloy. Specifically, we observe
the effect of varying the vertical domain extent ($\delta$) on tip selection,
by quantifying the dendrite tip velocity and curvature as a function of
$\delta$, and other process parameters. As $\delta$ decreases, we find that the
operating state of the dendrite tips becomes significantly affected by the
presence of finite boundaries. For particular boundary conditions, we observe a
switching of the growth state from 3-D to 2-D at very small $\delta$, in both
the pure material and alloy. We demonstrate that results from the alloy model
compare favorably with those from an experimental study investigating this
effect.
