Monotonically labelled Motzkin trees

Consider a rooted tree structure the nodes of which have been labelled monotonically by elements of { 1, 2, . . .,k}, which means that any sequence connecting the root of the tree with a leaf is weakly monotone . For fixed k asymptotic equivalents of the form CA gA °n; 2 (n --oo) to the numbers of such tree structures with n nodes are obtained for the family of extended unary-binary trees (i .e ., Motzkin trees) and for the family of extended unary-t-ary trees . Furthermore the numbers of (not extended) monotonically labelled binary and unary-binary trees are studied . For each of these families the asymptotic behaviour of qAas k -is determined . This is done by investigating a non-linear function sequence . The roots of the functions of this function sequence equal qA. . Thus one finds for instance qA .(log 2)/k (k -goo) for the family of extended unary-binary trees, and qA.-R/2k (k--oo) for the family of binary trees .