# The number of rooted ordered trees of max-out degree $k$

An ordered tree (also known as ordinal tree and plane tree) is a rooted tree in which the children of each node are ordered. It is known that the number of the ordered tree with $$n$$ edges is the $$n$$'the Catalan number $$C_n$$.

I am interested in the number of ordered trees in which the number of children of each node is at most $$k$$ (possibly less than $$k$$) for some small parameter $$k$$ (I am particularly interested in the case of $$k=4$$).

The special case of ordered trees with $$k=2$$ is called 0-1-2 trees (each node has 0, 1, or 2 children). It is known that the number of such trees with $$n$$ edges is the $$n$$'th Motzkin number $$M_n$$. For $$k>2$$, the number of ordered trees should lie between $$M_n$$ and $$C_n$$. We know $$\log_2(M_n) = 1.585 n - o(n)$$ and $$\log_2(C_n) = 2n - o(n)$$. I am interested in the constant behind $$n$$ for $$k=4$$.