Calculating unbalanced binary trees
It is a special caterpillar. 
Let $d(i)$ be the number of such trees with root $i$. We have $d(1)=1$; and $d(n)=\sum_{i=1}^{n-1}d(i)+1$, where "plus 1" is for the case of no children and $d(i)$ in sigma for the case that the left child is $i$. 
One can easily compute $d(n)$ in $O(n)$ time using a simple loop. Or it is not hard to solve the recurrence relation and get $d(i)=2^{i-1}$ for $i\geq 1$.