I know that the total number of possible $m$-ary tree with $n$ nodes is \begin{align} C_n&=\frac{1}{(m-1)n+1}{mn \choose n}, \end{align} which is the Catalan number. I want to know if I can get a tighter (in the binomial coefficient) bound for the total number of possible $m$-ary tree with $n$ nodes if I restrict the maximum height of the tree to $h$ -- i.e., trees can be of height $h$ or less.
Note: I am aware of the bijection between $m$-ary trees and $(m-1)$-dyck paths. From Brian M. Scott's answer, there does not seem to be a closed form for the number of bounded dyck paths. I was wondering if I could get a upper bound on that number that is tighter (in the binomial coefficient term) than the Catalan number?