Quantifying the virality of cascades is an important question across disciplines such as the transmission of disease, the spread of information and the diffusion of innovations. An appropriate virality metric should be able to disambiguate between a shallow, broadcast-like diffusion process and a deep, multi-generational branching process. Although several valuable works have been dedicated to this field, most of them fail to take the position of the diffusion source into consideration, which makes them fall into the trap of graph isomorphism and would result in imprecise estimation of cascade virality inevitably under certain circumstances. In this paper, we propose a root-aware approach to quantifying the virality of cascades with proper consideration of the root node in a diffusion tree. With applications on synthetic and empirical cascades, we show the properties and potential utility of the proposed virality measure. Based on preferential attachment mechanisms, we further introduce a model to mimic the growth of cascades. The proposed model enables the interpolation between broadcast and viral spreading during the growth of cascades. Through numerical simulations, we demonstrate the effectiveness of the proposed model in characterizing the virality of growing cascades. Our work contributes to the understanding of cascade virality and growth, and could offer practical implications in a range of policy domains including viral marketing, infectious disease and information diffusion.