Micronetworks of tree form

In a tree composed of I generations from the center, the number of end points on the outer surface of the network are given by 

These end points are assumed to be a-junctions that deform affinely to the strain. The internal junctions are all regarded as r-junctions. The total number of chains in the network is 

Starting from the central T-junction A, we can integrate over possible positions of the T-junctions one generation after another, and find the elastic free energy 

This agrees with the result of the tetrahedra model studied by Flory and Rehner [24] to investigate the effect of junction fluctuations in a micronetwork consisting of four chains which start from the corners of the tetrahedra and are connected by one central junction. For a macroscopic network, the prefactor becomes 

The micronetwork of tree form, however, has a significantly large fraction of 7-junctions relative to the total number of junctions. A correction to improve this deficiency is necessary for application to real networks [22]. 

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