This note derives asymptotic bounds for planted trees on n nodes, illustrating the application to eigenvalues.
This note derives asymptotic upper and lower bounds for the number of planted plane trees on n nodes assigned labels from the set {1,2,3...k} with the restriction that on any path from the root to a leaf, the labels must strictly decrease. We illustrate an application to calculating the largest eigenvalue of the adjacency matrix of a tree.
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Cheung et al. (2025) studied this question.
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