Walkthrough

Tree: $1-2-3-4-5$ (line). Range: $k_1=1$, $k_2=3$. Centroid is $3$. Distances from $3$: $\{2,1,0,1,2\}$ for nodes $\{1,2,3,4,5\}$. For node $1$ (distance $2$), look for distances in range $[1-2, 3-2] = [-1, 1]$. Since distances are non-negative, this is $[0, 1]$: nodes $3$ and $2,4$. That is $3$ pairs.

For node $2$ (distance $1$), range $[0, 2]$: all nodes work. Continue this process for all nodes, subtract same-subtree pairs, and sum. The arithmetic gets complex but the pattern is clear.