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We study level-set percolation of the Gaussian free field on the infinite d-regular tree for fixed d ≥ 3. Denoting by h the critical value, we obtain the following results: for h > h we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level h; for h < h we prove that the number of vertices connected over distance k above level h to a fixed vertex grows exponentially in k with positive probability. Furthermore, we show that thedoi:10.3929/ethz-b-000424327 fatcat:yjt2oq44yjbxplkzmbrg77a2lm