We consider a Bernoulli bond percolation on a random recursive tree of size n≫1, with supercritical parameter p n =1-c/lnn for some c>0 fixed. It is known that with high probability, there exists then a unique giant cluster of size G n ∼e -c n, and it follows from a recent result of J. Schweinsberg [Electron. J. Probab. 17, Paper No. 91, 50 p. (2012; Zbl 1284.92072)] that G n has non-Gaussian fluctuations. We provide an explanation of this by analyzing the effect of percolation on different phases of the growth of recursive trees. This alternative approach may be useful for studying percolation on other classes of trees, such as for instance regular trees.