A note on a result of R. Kemp on R-typly rooted planted plane trees

R. Kemp has shown that the average height of r-tuply rooted planted plane trees is $$\sqrt {\pi n} - \frac{1}{2}(r - 2) + O(\log (n)n^{1/2 - \varepsilon } ), \varepsilon > 0, n \to \infty ,$$ assuming that all such trees withn nodes are equally likely. We give a quite short proof of this result (with an error term ofO (1)).