On the maximum out-degree in random trees

00 1 + L: If some mild conditions are satisfied, then Y n "'-' c( cp( T ) / T t . n -3 /2 1 where TCP' (T) cp( T). We say that a node v in a rooted tree has out-degree k if v is incident with k edges that lead away from the root of Tn. Our object here is to the behaviour of the maximum ~ ~(Tn) of trees in F. After describing briefly in §2 the ;:"'A~vL .u,,,,,-,ufamilies of trees we shall be considering, we obtain bounds in §3 for Pr{~ < k} in terms of the function 00 rk(T) .z=CmT . Then in §4 we obtain certain inequalities involving the functions k rk( T), assuming henceforth that the coefficients are well-behaved. Our main result is in §5 where we show that if D( n) max { k : nr k( T) 2': I} then Pr{(l€)D(n) < ~(Tn) < (1 + €)D(n)} -t 1 as n -t 00. We consider the problem of estimating D(n) in §6; we find that D(n) rv logn/log(R/r) if cp(t) has a finite radius of convergence R while D(n) = o(logn) if cp(t) is an entire function. Finally, in §7 we consider some particular families of trees. For example, if 1>(t) (1t)-l and F is the family of plane trees then D(n) 1 + [log2 n]; and if 4>(t) = e and F is the family of rooted labelled trees then D(n) rv logn/loglogn. (\Ve remark that the problem of determining the behaviour of ~(Tn) for this last family was cC?nsidered earlier in [6].)