Summary: We study a reinforcement process on graphs \(G\) of bounded degree. The model involves a parameter \(\alpha>0\) governing the strength of reinforcement, and Poisson clock rates \(\lambda_v\) at the vertices \(v\) of the graph. When the Poisson clock at a vertex \(v\) rings, one of the edges incident to it is reinforced, with edge \(e\) being chosen with probability proportional to its current count (counts start from 1) raised to the power \(\alpha\). The main problem in such models is to describe the (random) subgraph \(\mathcal{E}_\infty\), consisting of edges that are reinforced infinitely often. In this paper, we focus on the finite connected components of \(\mathcal{E}_\infty\) in the strong reinforcement regime \((\alpha>1)\) with clock rates that are uniformly bounded above. We show here that when \(\alpha\) is sufficiently large, all connected components of \(\mathcal{E}_\infty\) are trees. When the firing rates \(\lambda_v\) are constant, we show that all components are trees of diameter at most 3 when \(\alpha\) is sufficiently large, and that there are infinitely many phase transitions as \(\alpha\downarrow 1\). For example, on the triangular lattice, increasingly large (odd) cycles appear as \(\alpha\downarrow 1\) (while on the square lattice no finite component of \(\mathcal{E}_\infty\) contains a cycle for any \(\alpha>1)\). Increasingly long paths and other structures appear in both lattices when taking \(\alpha\downarrow 1\). In the special case where \(G=\mathbb{Z}\) and \(\alpha>1\), all connected components of \(\mathcal{E}_\infty\) are finite and we show that the possible cluster sizes are non-monotone in \(\alpha\). We also present several open problems.