Relation between the number of leaves of a tree and its diameter

TL;DR

This paper determines exact formulas for the minimum number of leaves in a tree given its order and diameter, and vice versa, resolving previous bounds and extending understanding of tree structure relationships.

Contribution

It provides exact formulas for the minimum leaves in a tree based on order and diameter, and characterizes the minimum diameter for trees with a fixed number of leaves.

Findings

For even diameter, minimum leaves are eil 2(n-1)/d eil 2(n-2)/(d-1).

For odd diameter, minimum leaves are eil 2(n-2)/(d-1).

Minimum diameter for fixed leaves is 2, 2k+1, or 2k+2 depending on n and f.

Abstract

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 $kf+3\le n\le (k+1)f+1.$