Monotonically labelled Motzkin trees
This paper gives asymptotic values for the number of monotonically labelled extended Motzkin trees and the number of monotonically labelled (not extended) binary trees. A Motzkin tree (or unary-binary tree) is either a single leaf or is built up repeatedly by an internal node with either one or two ordered subtrees. A monotonic labelling has the property that the nodes of a k-node tree are labelled 1,2,...,k so that any sequence connecting the root of the tree with a leaf is weakly monotone.
scientific article; zbMATH DE number 4012638