Cascades represent an important phenomenon across various disciplines such as sociology, economy, psychology, political science, marketing, and epidemiology. An important property of cascades is their morphology, which encompasses the structure, shape, and size. However, cascade morphology has not been rigorously characterized and modeled in prior literature. In this paper, we propose a Multi-order Markov Model for the Morphology of Cascades ($M^4C$) that can represent and quantitatively characterize the morphology of cascades with arbitrary structures, shapes, and sizes. $M^4C$ can be used in a variety of applications to classify different types of cascades. To demonstrate this, we apply it to an unexplored but important problem in online social networks -- cascade size prediction. Our evaluations using real-world Twitter data show that $M^4C$ based cascade size prediction scheme outperforms the baseline scheme based on cascade graph features such as edge growth rate, degree distribution, clustering, and diameter. $M^4C$ based cascade size prediction scheme consistently achieves more than 90% classification accuracy under different experimental scenarios.