For the number of nodes to be divisible by 4, we can partition the nodes in 6 ways
The partitions can be made from a total of 10 nodes because 1 node will be occupied by the root.
Partitions:
Left Sub-tree = 4, Right Sub-tree = 6 OR Left Sub-tree = 6, Right Sub-tree = 4
Left Sub-tree = 2, Right Sub-tree = 8 OR Left Sub-tree = 8, Right Sub-tree = 2
Left Sub-tree = 0, Right Sub-tree = 10 OR Left Sub-tree = 10, Right Sub-tree = 0
Now, we have to calculate in how many ways can we construct an Unlabelled binary tree with the available number of nodes, and that is given by $(2n)Cn/n+1$
if you apply it here you will get:
$2*((8C4)/5)*((12C6)/7) + 2*((16C8)/9)*((4C2)/3) + 2*((20C10)/11)*$
This will evaluate to 43008