Calculating unbalanced binary trees Two-part question:


Is there any more specific name than "unbalanced binary tree" to describe the tree below? The distinguishing characteristics are:

i. The root node and left children must have either 0 or 2 children.

ii. Right children must have 0 children.

iii. All nodes must be positive non-zero integers.

iv. Any node with children must equal the sum of its children.
 
Given the value of the root node (in this case, 77), is there a standard algorithm to efficiently calculate all possible trees of this type?


i.e. (77, 76, 1), ..., (77, 1, 76), (77, (76, 75, 1), 1), etc.





         77
        /  \
       68   9
      /  \
     48   20
    /   \
   47    1
  /  \
 42   5
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 $i\geq 1$.