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The authors investigate the large-n behaviour of the number of lattice animals with n vertices having alpha cycles per vertex. They prove concavity and continuity properties for the corresponding growth constant and, in particular, show that lattice trees are exponentially scarce in the set of lattice animals. They also consider the corresponding generating function and prove a number of theorems which bound it and set other limits on its possible behaviour.
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Madras et al. (1988) studied this question.
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