In this paper,we mainly study the total number of leaves of Motzkin trees with n edges.By using symbolic method and bijection respectively,we prove that the total number of leaves of Motzkin trees with n edges is equal to the number of free Motzkin paths with length n.Using this bijection we also get the total number of leaves of full binary trees with n internal nodes,and give a new proof that the number of Dyck paths with semilength n and k leaves is the Narayana number.
Motzkin treesfree Motzkin pathssymbolic methodNarayana number
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