Creating a closed parent-child network structure

Question

I am new to this forum, and I am hoping that there is a mathematical solution to this situation. I am trying to form a network structure of parent and children nodes, but with some conditions. Following is a description:

Assume a finite number of nodes (Let's say 50 or take any finite one number. In an application of this structure, the real number might be very high). Now a closed structure of these nodes is to be constructed such that

1. Every node always has a single parent (there can be multiple parents in the network)
2. Every node always has a minimum of one child, there is no limit to the maximum number of children. In short, every node is a parent to atleast one node. If not every node, then a maximum number of nodes have atleast one child.

Here is an example of a decentralized network. But it is an open network with end nodes without children, whereas I am looking at possibly a closed network with minimum or zero "childless" nodes.

Can such a structure exist? Any thoughts would be really appreciated.

Thanks!