Abstract
A leaf of a NC-tree is called a left leaf if it has a label less than that of its parent, and it is called a right leaf otherwise. We enumerate NC-trees with given number of left leaves and right leaves. As an application, the combinatorial interpretation of three identities is obtained. By extending a well-known involution on plane trees to NC-trees, we derive that the two parameters, the number of left leaves and the number of non-root proper vertices, have a symmetric joint distribution in the NC-trees. Finally, a bijection between right leaves and NC-forests is also presented.
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