How many number of different binary trees are possible for a given postorder (or preorder) traversal
Every binary tree (with the right number of nodes) has exactly one labelling that satisfies a given postorder labelling. So you need to find the number of binary trees. That is the famous Catalan number $C_n = \frac 1{n+1}{2n \choose n}$. Sequence A000108 in Sloane's OIES.

It has a nice recurrence, based on the fact that a tree with $n+1$ nodes has a root and the remaining nodes are distributed into two subtrees with $i$ and $n-i$ nodes respectively: $C_{n+1}=\sum_{i=0}^n C_i\,C_{n-i} $. Such a type of recurrence is called a convolution.

In fact, you have discovered this recurrence yourself! Here are your own numbers, in a slightly changed order:

$1:4:0=14 = 14 \cdot 1$

$1:3:1=5 = 5 \cdot 1$

$1:2:2=4 = 2 \cdot 2$

$1:1:3=5 = 1 \cdot 5$

$1:0:4=14 = 1 \cdot 14$