|We want to design a network consisting of n identical Worker nodes and one Server node.
The network must be connected
and must not contain any cycles. We also require that each Worker node must be connected to an
odd number of other nodes. How many distinct networks are there?
Here is a listing of all the distinct networks containing exactly 5 worker nodes:
W W W W W W
| \ | | | /
W---S----W S---W S---W S---W
| \ / | | | \
W W W W W---W---W W W
Two configurations G and G' are not distinct if there is a 1-1 mapping between their nodes such that the server in G
maps to the server in G', and such that for every pair of nodes in G the pair that they are mapped to in G' is connected
if and only if the pair in G is connected. Note that two configurations are not distinct if they have the same connection pattern, even if they are different geometrically as displayed in the plane. For example, these
two configurations with 8 worker nodes are not distinct:
W W W W W
| | | | |
| | | | |
W W W W---W---W W
Given n, return the number of distinct networks that can
be constructed with exactly n worker nodes.