What's the number of distinct binary trees possible with n distinct nodes?
For a binary tree with n nodes, the number of edges is n-1.So, this problem can be reduced to the number of ways in which we can make n-1 edges from n vertices. An edge can be made either as a left child of a vertex or as a right child. Hence, for n vertices, we have 2n possibilities for the first edge chosen, 2n-1 for the second and so on. Thus for n-1 the total number of ways
= $2n * (2n-1) * (2n-2) * .... * (2n - (n - 2))$
= $2n * (2n-1) * (2n-2) * .... * (n + 2)$
=$ \frac{(2n)!} { (n+1)!}$
What's the number of distinct binary trees possible with n distinct nodes?
For a binary tree with n nodes, the number of edges is n-1.So, this problem can be reduced to the number of ways in which we can make n-1 edges from n vertices. An edge can be made either as a left child of a vertex or as a right child. Hence, for n vertices, we have 2n possibilities for the first edge chosen, 2n-1 for the second and so on. Thus for n-1 the total number of ways
= $2n * (2n-1) * (2n-2) * .... * (2n - (n - 2))$
= $2n * (2n-1) * (2n-2) * .... * (n + 2)$
=$ \frac{(2n)!} { (n+1)!}$
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