Two proofs of a frequently rediscovered combinatorial lemma are presented. Using the lemma, a combinatorial proof is given that the average height of an ordered (plane-planted) tree is approximately twice the average node (vertex) level. © 1990, Academic Press Limited. All rights reserved.
Dershowitz, N., & Zaks, S. (1990). The Cycle Lemma and Some Applications. European Journal of Combinatorics, 11(1), 35–40. https://doi.org/10.1016/S0195-6698(13)80053-4