Leaf-to-leaf path length conjecture for trees with a given degree sequence

Let $T$ be a tree with degree sequence $s$, and suppose that $T$ has no vertex of degree $2$. Let $lp(T)$ denote the number of different lengths of leaf-to-leaf paths in $T$, and let ${\rm rad}(s)$ denote the minimum radius of a tree with degree sequence $s$.

Leaf-to-leaf path length conjecture.

$$lp(T)\geq {\rm rad}(s)-O(1).$$

This conjecture seeks a lower bound on the number of distinct leaf-to-leaf path lengths determined by the degree sequence, strengthening bounds in terms of the number of leaves and maximum degree. The paper proves a weaker bound with a logarithmic error term, namely $lp(T)\geq {\rm rad}(s)-\log_2({\rm rad}(s))$, while the stated bounded-error improvement remains open.