analysis of exponential trees via contraction method

The exponential trees model certain aspects of fast-growing social networks.in exponentially growing trees, each node independently attracts a new nodewith probability p, or not to attract with probability 1 − p. recently, the size of thesetrees, number of leaves, profile and total path length are investigated. here, we studythe asymptotic behavior of the number of protected nodes, nodes that are not leaf andnot all of their children are leaves. we also give some characterizations of the limit lawof the total path length of protected nodes via contraction method.