For $\displaystyle \Delta \geq 3 $ , consider all trees where each vertex has degree $\displaystyle \Delta $ or less, and where the longest path is of length k or less. Show that the maximum number $\displaystyle l_{max} $ of leaves among such trees satisfies the following:

$\displaystyle l_{max} \leq \Delta (\Delta - 1)^{k/2-1} $