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In humid savannas, the transition from juvenile to mature tree sizes is thought to be a major demographic bottleneck because smaller plants are topkilled by frequent fires. Species with the highest net rates of sapling growth should dominate the tree component of savannas by reaching fire-proof sizes, ‘escape size’, more rapidly than competitors. However, tests of this prediction have failed to explain eucalypt dominance in Australian savannas as eucalypt mean growth rates are low. We tested the escape hypothesis directly by recording the number and identity of tagged individuals reaching escape size in a long-term study. Results were consistent with the escape hypothesis with greater than six times more eucalypts reaching escape height than non-eucalypts in frequently burnt savannas. The pattern was reversed where fires were excluded for 5 years, with more non-eucalypts emerging than eucalypts. We conclude that mean growth rates alone are a poor predictor of the rates at which juvenile savanna trees transition to mature tree size. Only the fastest growing individuals make the transition to mature trees whereas mean values include many suppressed individuals. Measures of maximum growth rates will provide more robust estimators of demographically important effects on mature tree cover. The mechanistic basis for the remarkable ability of juvenile eucalypts to escape frequent fires is not yet understood.