Trees have been used for counting chemical structures of non-isomorphic forms. Graph enumeration can be either isomorphic or non-isomorphic. Isomorphic tree enumeration arises in the context of listing all isomorphic subtrees of a given graph whereas non-isomorphic enumeration lists all unique trees that satisfy given constraints. We study the chemical structures that grow like a tree. They grow from a center, layer by layer, and expand in a star like manner where all vertices at layer $l$ are $l$ hops from the root. We presume that the tree is rooted at the center where the center can be either an edge or a vertex. A subtree rooted at an immediate child of the root is called as a branch. A cluster in a particular layer $j$ is the number of descendants of a particular branch in layer $j$ . Each layer is characterized by its largest cluster. We count the chemical structures, i.e. star-shaped polymers that meet the constraints that are defined in terms of the largest cluster. We define the notion of balance in trees and seek to count the number of balanced trees (i.e. polymers) when all trees (polymers) are presumed to be possible.