The crystallization of polymeric and metallic materials normally occurs under conditions far from equilibrium, leading to patterns that grow as propagating waves into the surrounding unstable fluid medium. The Mullins-Sekerka instability causes these wave fronts to break up into dendritic arms, and we anticipate that the normal modes of the dendrite tips have a significant influence on pattern growth. To check this possibility, we focus on the dendritic growth of polyethylene oxide in a thin-film geometry. This crystalline polymer is mixed with an amorphous polymer (polymethyl-methacrylate) to "tune" the morphology and clay was added to nucleate the crystallization. The tips of the main dendrite trunks pulsate during growth and the sidebranches, which grow orthogonally to the trunk, pulsate out of phase so that the tip dynamics is governed by a limit cycle. The pulsation period P increases sharply with decreasing film thickness L and then vanishes below a critical value L(c) approximately 80 nm. A change of dendrite morphology accompanies this transition.