Problems: Prove that the number of plane trees with $(n+1)$ vertices is $n-th$ Catalan number $C_n$.
For this problem: I have found a solution in a post on Mathstack Prove that number of non-isomorphic ordered tree with 'n' vertices is nth catalan number.
However, I wanna try another approach using the same argument as I used for the case full binary tree including $n+1$ internal nodes: "A tree $T$ with $n+1$ nodes has a root with two subtrees $t_1$ and $t_2$ have $n$ internal nodes in total. So there is $T_iT_{n-i}$ ways to form the two such subtrees". Taking the sum of $i$ from $0$ to $n$, we obtain the Catalan recurrence.
But for the plane tree, using the same argument seems to be unfeasible (figure below)
I wonder if there is any simple way to approach this problem. I hope everyone could give me some hints.