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The central limit theorems (CLT's) for some quantities related to Bernoulli percolation problems are studied by many authors. For Bernoulli bond percolation problem on Zd (d 2), Zhang(3) proved a CLT for the number of open clusters in a nite box for all p 2 (0; 1), that is, including the case p = pc(Zd). Using his method together with the ergodic theorem, Penrose(2) proved a general CLT which can be applied to several models. We study limit theorems for percolation problems on pre-fractal graphs and trees, where the ergodic theorem is not available. For a class of Sierpi nski carpet lattices, considered by Kumagai(1), we obtain some limit theorems for the number of percolation clusters. We can also prove a CLT for the number of percolation clusters on trees of bounded degree.

Remarks on Central Limit Theorems for the Number of Percolation Clusters