Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $d.$ In 1975 Lesniak gave the lower bound
  $B(n,d)=\lceil 2(n-1)/d\rceil$ 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 $L(21,3)=19.$ We prove that for $d\ge 2,$
  $ L(n,d)=\left\lceil \frac{2(n-1)}{d}\right\rceil$ if $d$ is even and
$L(n,d)=\left\lceil \frac{2(n-2)}{d-1}\right\rceil$ if $d$ is odd.
  The converse problem is also considered. Let $D(n,f)$ be the minimum possible
diameter of a tree of order $n$ with exactly $f$ leaves.
  We prove that $D(n,f)=2$ if $n=f+1,$ $D(n,f)=2k+1$ if $n=kf+2,$ and
$D(n,f)=2k+2$ if $kf+3\le n\le (k+1)f+1.$

36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph