The relation between the number of leaves of a tree and its diameter

Let L ( n, d ) denote the minimum possible number of leaves in a tree of order n and diameter d . Lesniak (1975) gave the lower bound B ( n,d ) = ⌈2( n − 1)/ d ⌉ for L ( n,d ). When d is even, B ( n,d ) = L ( n,d ). But when d is odd, B ( n,d ) is smaller than L ( n,d ) in general. For example, B (21, 3) = 14 while L (21, 3) = 19. In this note, we determine L ( n, d ) using new ideas. We also consider the converse problem and determine the minimum possible diameter of a tree with given order and number of leaves.