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Internet worm infection continues to be one of top security threats and has been widely used by botnets to recruit new bots. In this work, we attempt to quantify the infection ability of individual hosts and reveal the key characteristics of the underlying topology formed by worm infection, i.e., the number of children and the generation of the worm infection family tree. Specifically, we first apply probabilistic modeling methods and a sequential growth model to analyze the infection tree of a wide class of worms. Through both mathematical analysis and simulation, we find that the number of children has asymptotically a geometric distribution with parameter 0.5. As a result, on average half of infected hosts never compromise any vulnerable host, over 98% of infected hosts have no more than five children, and a small portion of infected hosts have a large number of children. We also discover that the generation follows closely a Poisson distribution and the average path length of the worm infection family tree increases approximately logarithmically with the total number of infected hosts. Next, we study the infection structure of localized-scanning and permutation-scanning worms through simulation and surprisingly find that the above observations also apply to these worms. Finally, we apply our findings to evaluate bot assessment strategies for forensic analysis after a worm tree has been formed.