Abstract—Closed-form solutions for the 50 % delay, rise time, overshoots, and settling time of signals in an tree are presented. These solutions have the same accuracy characteristics of the Elmore delay for trees and preserves the simplicity and recursive characteristics of the Elmore delay. Specifically, the complexity of calculating the time domain responses at all the nodes of an tree is linearly proportional to the number of branches in the tree and the solutions are always stable. The closed-form expressions introduced here consider all damping conditions of an circuit including the underdamped response, which is not considered by the Elmore delay due to the nonmonotone nature of the response. The continuous analytical nature of the solutions makes these expressions suitable for design methodologies and optimization techniques. Also, the solutions have significantly improved accuracy as compared to the Elmore delay for an overdamped response. The solutions introduced here for trees can be practically used for the same purposes that the Elmore delay is used for trees
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